1.
Euler diagram
–
An Euler diagram is a diagrammatic means of representing sets and their relationships. Typically they involve overlapping shapes, and may be scaled, such that the area of the shape is proportional to the number of elements it contains and they are particularly useful for explaining complex hierarchies and overlapping definitions. They are often confused with the Venn diagrams, unlike Venn diagrams which show all possible relations between different sets, the Euler diagram shows only relevant relationships. The first use of Eulerian circles is commonly attributed to Swiss mathematician Leonhard Euler, in the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since then, they have also adopted by other curriculum fields such as reading as well as organizations. Euler diagrams consist of simple closed shapes in a two dimensional plane that depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets, there are only 3 possible relationships between any 2 sets, completely inclusive, partially inclusive, and exclusive. This is also referred to as containment, overlap or neither or, especially in mathematics, it may be referred to as subset, intersection, curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements, a curve that is contained completely within the interior zone of another represents a subset of it. Venn diagrams are a more form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, when the number of sets grows beyond 3 a Venn diagram becomes visually complex, especially compared to the corresponding Euler diagram. The difference between Euler and Venn diagrams can be seen in the following example, the Venn diagram, which uses the same categories of Animal, Mineral, and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region, Euler diagrams represent emptiness either by shading or by the absence of a region. Often a set of conditions are imposed, these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, in the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations, some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs
Euler diagram
–
Photo of page from Hamilton's 1860 "Lectures" page 180. (Click on it, up to two times, to enlarge). The symbolism A, E, I, and O refer to the categorical statements that can occur in a
syllogism. The small text to the left says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise."
Euler diagram
–
An Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is disjoint (has no members in common) with "animals"
Euler diagram
Euler diagram
–
Both the Veitch and Karnaugh diagrams show all the
minterms, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables x, y, and z are per Venn's example.
2.
Mathematics
–
Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
Mathematics
–
Euclid (holding
calipers), Greek mathematician, 3rd century BC, as imagined by
Raphael in this detail from
The School of Athens.
Mathematics
–
Greek mathematician
Pythagoras (c. 570 – c. 495 BC), commonly credited with discovering the
Pythagorean theorem
Mathematics
–
Leonardo Fibonacci, the
Italian mathematician who established the Hindu–Arabic numeral system to the Western World
Mathematics
–
Carl Friedrich Gauss, known as the prince of mathematicians
3.
Set theory
–
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined
Set theory
–
Georg Cantor
Set theory
–
A
Venn diagram illustrating the
intersection of two
sets.
4.
Set (mathematics)
–
In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
Set (mathematics)
–
A set of polygons in a
Venn diagram
5.
Element (mathematics)
–
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1,2,3 and 4, sets of elements of A, for example, are subsets of A. For example, consider the set B =, the elements of B are not 1,2,3, and 4. Rather, there are three elements of B, namely the numbers 1 and 2, and the set. The elements of a set can be anything, for example, C =, is the set whose elements are the colors red, green and blue. The relation is an element of, also called set membership, is denoted by the symbol ∈, writing x ∈ A means that x is an element of A. Equivalent expressions are x is a member of A, x belongs to A, x is in A and x lies in A, another possible notation for the same relation is A ∋ x, meaning A contains x, though it is used less often. The negation of set membership is denoted by the symbol ∉, writing x ∉ A means that x is not an element of A. The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita, here he wrote on page X, Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b. which means The symbol ϵ means is, so a ϵ b is read as a is a b. The symbol itself is a stylized lowercase Greek letter epsilon, the first letter of the word ἐστί, the Unicode characters for these symbols are U+2208, U+220B and U+2209. The equivalent LaTeX commands are \in, \ni and \notin, mathematica has commands \ and \. The number of elements in a set is a property known as cardinality, informally. In the above examples the cardinality of the set A is 4, an infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets, an example of an infinite set is the set of positive integers =. Using the sets defined above, namely A =, B = and C =,2 ∈ A ∈ B3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite, the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY, Springer-Verlag, ISBN 0-387-90092-6 - Naive means that it is not fully axiomatized, not that it is silly or easy. Jech, Thomas, Set Theory, Stanford Encyclopedia of Philosophy Suppes, Patrick, Axiomatic Set Theory, NY, Dover Publications, Inc
Element (mathematics)
–
First usage of the symbol ϵ in the work Arithmetices principia nova methodo exposita by
Giuseppe Peano.
6.
Partial order
–
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for pairs of elements in the set. The word partial in the partial order or partially ordered set is used as an indication that not every pair of elements need be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset, Partial orders thus generalize total orders, in which every pair is comparable. To be an order, a binary relation must be reflexive, antisymmetric. One familiar example of an ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, a poset can be visualized through its Hasse diagram, which depicts the ordering relation. A partial order is a binary relation ≤ over a set P satisfying particular axioms which are discussed below, when a ≤ b, we say that a is related to b. The axioms for a partial order state that the relation ≤ is reflexive, antisymmetric. That is, for all a, b, and c in P, it must satisfy, in other words, a partial order is an antisymmetric preorder. A set with an order is called a partially ordered set. The term ordered set is also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as ordered sets, for a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. In the figure on top-right, e. g. and are comparable, while and are not, a partial order under which every pair of elements is comparable is called a total order or linear order, a totally ordered set is also called a chain. A subset of a poset in which no two elements are comparable is called an antichain. A more concise definition will be given using the strict order corresponding to ≤. For example, is covered by in the figure. Standard examples of posets arising in mathematics include, The real numbers ordered by the standard less-than-or-equal relation ≤, the set of subsets of a given set ordered by inclusion
Partial order
–
Partially ordered set of
set of all subsets of a six-element set {a, b, c, d, e, f}, ordered by the subset relation.
Partial order
–
The
Hasse diagram of the
set of all subsets of a three-element set {x, y, z}, ordered by inclusion. Sets on the same horizontal level don't share a precedence relationship. Some other pairs, such as {x} and {y,z}, do not either.
7.
Boolean algebra (structure)
–
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets and it is also a special case of a De Morgan algebra and a Kleene algebra. The term Boolean algebra honors George Boole, a self-educated English mathematician, booles formulation differs from that described above in some important respects. For example, conjunction and disjunction in Boole were not a pair of operations. Boolean algebra emerged in the 1860s, in papers written by William Jevons, the first systematic presentation of Boolean algebra and distributive lattices is owed to the 1890 Vorlesungen of Ernst Schröder. The first extensive treatment of Boolean algebra in English is A. N. Whiteheads 1898 Universal Algebra, Boolean algebra as an axiomatic algebraic structure in the modern axiomatic sense begins with a 1904 paper by Edward V. Huntington. Boolean algebra came of age as serious mathematics with the work of Marshall Stone in the 1930s, and with Garrett Birkhoffs 1940 Lattice Theory. In the 1960s, Paul Cohen, Dana Scott, and others found deep new results in mathematical logic and axiomatic set theory using offshoots of Boolean algebra, namely forcing, a Boolean algebra with only one element is called a trivial Boolean algebra or a degenerate Boolean algebra. It follows from the last three pairs of axioms above, or from the axiom, that a = b ∧ a if. The relation ≤ defined by a ≤ b if these equivalent conditions hold, is an order with least element 0. The meet a ∧ b and the join a ∨ b of two elements coincide with their infimum and supremum, respectively, with respect to ≤, the first four pairs of axioms constitute a definition of a bounded lattice. It follows from the first five pairs of axioms that any complement is unique, the set of axioms is self-dual in the sense that if one exchanges ∨ with ∧ and 0 with 1 in an axiom, the result is again an axiom. Therefore, by applying this operation to a Boolean algebra, one obtains another Boolean algebra with the same elements, furthermore, every possible input-output behavior can be modeled by a suitable Boolean expression. The smallest element 0 is the empty set and the largest element 1 is the set S itself, starting with the propositional calculus with κ sentence symbols, form the Lindenbaum algebra. This construction yields a Boolean algebra and it is in fact the free Boolean algebra on κ generators. A truth assignment in propositional calculus is then a Boolean algebra homomorphism from this algebra to the two-element Boolean algebra, interval algebras are useful in the study of Lindenbaum-Tarski algebras, every countable Boolean algebra is isomorphic to an interval algebra. For any natural n, the set of all positive divisors of n, defining a≤b if a divides b
Boolean algebra (structure)
–
Boolean lattice of subsets
8.
Power set
–
In mathematics, the power set of any set S is the set of all subsets of S, including the empty set and S itself. The power set of a set S is variously denoted as P, ℘, P, ℙ, or, in axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set. Any subset of P is called a family of sets over S, if S is the set, then the subsets of S are, and hence the power set of S is. If S is a set with |S| = n elements. This fact, which is the motivation for the notation 2S, may be demonstrated simply as follows, First and we write any subset of S in the format where γi,1 ≤ i ≤ n, can take the value of 0 or 1. If γi =1, the element of S is in the subset, otherwise. Clearly the number of subsets that can be constructed this way is 2n as γi ∈. Cantors diagonal argument shows that the set of a set always has strictly higher cardinality than the set itself. In particular, Cantors theorem shows that the set of a countably infinite set is uncountably infinite. The power set of the set of numbers can be put in a one-to-one correspondence with the set of real numbers. The power set of a set S, together with the operations of union, intersection, in fact, one can show that any finite Boolean algebra is isomorphic to the Boolean algebra of the power set of a finite set. For infinite Boolean algebras this is no true, but every infinite Boolean algebra can be represented as a subalgebra of a power set Boolean algebra. The power set of a set S forms a group when considered with the operation of symmetric difference. It can hence be shown that the power set considered together with both of these forms a Boolean ring. In set theory, XY is the set of all functions from Y to X, as 2 can be defined as, 2S is the set of all functions from S to. Hence 2S and P could be considered identical set-theoretically and this notion can be applied to the example above in which S = to see the isomorphism with the binary numbers from 0 to 2n −1 with n being the number of elements in the set. In S, a 1 in the corresponding to the location in the set indicates the presence of the element. The number of subsets with k elements in the set of a set with n elements is given by the number of combinations, C
Power set
–
The elements of the power set of the set { x, y, z }
ordered in respect to
inclusion.
9.
Cardinality
–
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = contains 3 elements, there are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a bar on each side, this is the same notation as absolute value. Alternatively, the cardinality of a set A may be denoted by n, A, card, while the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets. Two sets A and B have the same cardinality if there exists a bijection, that is, such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B, for example, the set E = of non-negative even numbers has the same cardinality as the set N = of natural numbers, since the function f = 2n is a bijection from N to E. A has cardinality less than or equal to the cardinality of B if there exists a function from A into B. A has cardinality less than the cardinality of B if there is an injective function. If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |, the axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A, B. That is, the cardinality of a set was not defined as an object itself. However, such an object can be defined as follows, the relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all sets which have the same cardinality as A. There are two ways to define the cardinality of a set, The cardinality of a set A is defined as its class under equinumerosity. A representative set is designated for each equivalence class, the most common choice is the initial ordinal in that class. This is usually taken as the definition of number in axiomatic set theory. Assuming AC, the cardinalities of the sets are denoted ℵ0 < ℵ1 < ℵ2 < …. For each ordinal α, ℵ α +1 is the least cardinal number greater than ℵ α
Cardinality
–
Bijective function from N to E. Although E is a proper subset of N, both sets have the same cardinality.
10.
Inequality (mathematics)
–
In mathematics, an inequality is a relation that holds between two values when they are different. The notation a ≠ b means that a is not equal to b and it does not say that one is greater than the other, or even that they can be compared in size. If the values in question are elements of a set, such as the integers or the real numbers. The notation a < b means that a is less than b, the notation a > b means that a is greater than b. In either case, a is not equal to b and these relations are known as strict inequalities. The notation a < b may also be read as a is less than b. The notation a ≥ b means that a is greater than or equal to b, not less than can also be represented by the symbol for less than bisected by a vertical line, not. In engineering sciences, a formal use of the notation is to state that one quantity is much greater than another. The notation a ≪ b means that a is less than b. The notation a ≫ b means that a is greater than b. Inequalities are governed by the following properties, all of these properties also hold if all of the non-strict inequalities are replaced by their corresponding strict inequalities and monotonic functions are limited to strictly monotonic functions. The transitive property of inequality states, For any real numbers a, b, c, If a ≥ b and b ≥ c, If a ≤ b and b ≤ c, then a ≤ c. If either of the premises is an inequality, then the conclusion is a strict inequality. E. g. if a ≥ b and b > c, then a > c An equality is of course a special case of a non-strict inequality. E. g. if a = b and b > c, then a > c The relations ≤ and ≥ are each others converse, For any real numbers a and b, If a ≤ b, then b ≥ a. If a ≥ b, then a + c ≥ b + c, If a ≤ b and c >0, then ac ≤ bc and a/c ≤ b/c. If c is negative, then multiplying or dividing by c inverts the inequality, If a ≥ b and c <0, then ac ≤ bc, If a ≤ b and c <0, then ac ≥ bc and a/c ≥ b/c. More generally, this applies for a field, see below
Inequality (mathematics)
–
The
feasible regions of
linear programming are defined by a set of inequalities.
11.
Empty set
–
In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
Empty set
–
The empty set is the set containing no elements.
12.
Prime number
–
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
Prime number
–
The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.
13.
Natural number
–
In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory
Natural number
–
The
Ishango bone (on exhibition at the
Royal Belgian Institute of Natural Sciences) is believed to have been used 20,000 years ago for natural number arithmetic.
Natural number
–
Natural numbers can be used for counting (one
apple, two apples, three apples, …)
14.
Rational number
–
In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all numbers, often referred to as the rationals, is usually denoted by a boldface Q, it was thus denoted in 1895 by Giuseppe Peano after quoziente. The decimal expansion of a rational number always either terminates after a number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. In abstract algebra, the numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d
Rational number
–
A diagram showing a representation of the equivalent classes of pairs of integers
15.
Line segment
–
In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while a line segment excludes both endpoints, a half-open line segment includes exactly one of the endpoints. Examples of line include the sides of a triangle or square. More generally, when both of the end points are vertices of a polygon or polyhedron, the line segment is either an edge if they are adjacent vertices. When the end points both lie on a such as a circle, a line segment is called a chord. Sometimes one needs to distinguish between open and closed line segments, thus, the line segment can be expressed as a convex combination of the segments two end points. In geometry, it is defined that a point B is between two other points A and C, if the distance AB added to the distance BC is equal to the distance AC. Thus in R2 the line segment with endpoints A = and C = is the collection of points. A line segment is a connected, non-empty set, if V is a topological vector space, then a closed line segment is a closed set in V. However, an open line segment is an open set in V if and only if V is one-dimensional. More generally than above, the concept of a segment can be defined in an ordered geometry. A pair of segments can be any one of the following, intersecting, parallel, skew. The last possibility is a way that line segments differ from lines, in an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or else be defined in terms of an isometry of a line. Segments play an important role in other theories, for example, a set is convex if the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of sets to the analysis of a line segment. The Segment Addition Postulate can be used to add congruent segment or segments with equal lengths and consequently substitute other segments into another statement to make segments congruent. A line segment can be viewed as a case of an ellipse in which the semiminor axis goes to zero, the foci go to the endpoints. A complete orbit of this ellipse traverses the line segment twice, as a degenerate orbit this is a radial elliptic trajectory. In addition to appearing as the edges and diagonals of polygons and polyhedra, some very frequently considered segments in a triangle include the three altitudes, the three medians, the perpendicular bisectors of the sides, and the internal angle bisectors
Line segment
–
historical image – create a line segment (1699)
16.
Line (mathematics)
–
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f
Line (mathematics)
–
The red and blue lines on this graph have the same
slope (gradient); the red and green lines have the same
y-intercept (cross the
y-axis at the same place).
17.
Real number
–
In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing
Real number
–
A symbol of the set of real numbers (ℝ)
18.
Isomorphic
–
In mathematics, an isomorphism is a homomorphism or morphism that admits an inverse. Two mathematical objects are isomorphic if an isomorphism exists between them, an automorphism is an isomorphism whose source and target coincide. For most algebraic structures, including groups and rings, a homomorphism is an isomorphism if, in topology, where the morphisms are continuous functions, isomorphisms are also called homeomorphisms or bicontinuous functions. In mathematical analysis, where the morphisms are functions, isomorphisms are also called diffeomorphisms. A canonical isomorphism is a map that is an isomorphism. Two objects are said to be isomorphic if there is a canonical isomorphism between them. Isomorphisms are formalized using category theory, let R + be the multiplicative group of positive real numbers, and let R be the additive group of real numbers. The logarithm function log, R + → R satisfies log = log x + log y for all x, y ∈ R +, so it is a group homomorphism. The exponential function exp, R → R + satisfies exp = for all x, y ∈ R, the identities log exp x = x and exp log y = y show that log and exp are inverses of each other. Since log is a homomorphism that has an inverse that is also a homomorphism, because log is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to real numbers using a ruler. Consider the group, the integers from 0 to 5 with addition modulo 6 and these structures are isomorphic under addition, if you identify them using the following scheme, ↦0 ↦1 ↦2 ↦3 ↦4 ↦5 or in general ↦ mod 6. For example, + =, which translates in the system as 1 +3 =4. Even though these two groups look different in that the sets contain different elements, they are indeed isomorphic, more generally, the direct product of two cyclic groups Z m and Z n is isomorphic to if and only if m and n are coprime. For example, R is an ordering ≤ and S an ordering ⊑, such an isomorphism is called an order isomorphism or an isotone isomorphism. If X = Y, then this is a relation-preserving automorphism, in a concrete category, such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories, an isomorphism is the same as a homomorphism which is bijective on underlying sets, in abstract algebra, two basic isomorphisms are defined, Group isomorphism, an isomorphism between groups Ring isomorphism, an isomorphism between rings. Just as the automorphisms of an algebraic structure form a group, letting a particular isomorphism identify the two structures turns this heap into a group
Isomorphic
–
The group of fifth
roots of unity under multiplication is isomorphic to the group of rotations of the regular pentagon under composition.
19.
Ordinal number
–
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence
Ordinal number
–
Representation of the ordinal numbers up to ω ω. Each turn of the spiral represents one power of ω
20.
Cartesian product
–
In Set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B, products can be specified using set-builder notation, e. g. A table can be created by taking the Cartesian product of a set of rows, If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More generally, a Cartesian product of n sets, also known as an n-fold Cartesian product, can be represented by an array of n dimensions, an ordered pair is a 2-tuple or couple. The Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, an illustrative example is the standard 52-card deck. The standard playing card ranks form a 13-element set, the card suits form a four-element set. The Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form, both sets are distinct, even disjoint. The main historical example is the Cartesian plane in analytic geometry, usually, such a pairs first and second components are called its x and y coordinates, respectively, cf. picture. The set of all such pairs is thus assigned to the set of all points in the plane, a formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair. The most common definition of ordered pairs, the Kuratowski definition, is =, note that, under this definition, X × Y ⊆ P, where P represents the power set. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, union, power set, let A, B, C, and D be sets. × C ≠ A × If for example A =, then × A = ≠ = A ×, the Cartesian product behaves nicely with respect to intersections, cf. left picture. × = ∩ In most cases the above statement is not true if we replace intersection with union, cf. middle picture. Other properties related with subsets are, if A ⊆ B then A × C ⊆ B × C, the cardinality of a set is the number of elements of the set. For example, defining two sets, A = and B =, both set A and set B consist of two elements each. Their Cartesian product, written as A × B, results in a new set which has the following elements, each element of A is paired with each element of B. Each pair makes up one element of the output set, the number of values in each element of the resulting set is equal to the number of sets whose cartesian product is being taken,2 in this case
Cartesian product
–
Standard 52-card deck
Cartesian product
–
Cartesian product of the sets and
21.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
International Standard Book Number
–
A 13-digit ISBN, 978-3-16-148410-0, as represented by an
EAN-13 bar code
22.
McGraw-Hill
–
The company also provides reference and trade publications for the medical, business, and engineering professions. McGraw-Hill Education currently operates in 28 countries, has more than 4,800 employees globally and this shift has accelerated in recent years with an increased focus on developing adaptive learning systems that enable classroom teaching to come closer to a one-to-one student-teacher interaction. These systems allow personalized learning by assessing each students skill level, McGraw-Hill Education provides digital products and services to over 11 million users. In 2013, the acquired the ALEKS Corporation and after acquiring 20 percent equity stake in Area9 ApS went on to acquire the company. In 2015 MHE opened a new R&D office in Boston’s innovation district, in September 2016 the company acquired adaptive learning technology and content provider Redbird Learning. The company currently offers over 1,500 adaptive products in higher education, McGraw-Hill Education traces its history back to 1888 when James H. McGraw, co-founder of the company, purchased the American Journal of Railway Appliances. He continued to add further publications, eventually establishing The McGraw Publishing Company in 1899 and his co-founder, John A. Hill, had also produced several technical and trade publications and in 1902 formed his own business, The Hill Publishing Company. In 1909 both men agreed upon an alliance and combined the book departments of their publishing companies into The McGraw-Hill Book Company, John Hill served as President, with James McGraw as Vice-President. 1917 saw the merger of the parts of each business into The McGraw-Hill Publishing Company. In 1986, McGraw-Hill bought out competitor The Economy Company, then the nations largest publisher of educational material, the buyout made McGraw-Hill the largest educational publisher in the U. S. In 1979 McGraw-Hill Publishing Company purchased Byte from its owner/publisher Virginia Williamson who then became a vice-president of McGraw-Hill, McGraw-Hill Publishing Company, Inc became The McGraw-Hill Companies in 1995, as part of a corporate identity rebranding. In 2007, The McGraw-Hill Companies launched a student study network. This offering gave McGraw-Hill an opportunity to connect directly with its end users, the site closed on April 29,2012. On November 26,2012, The McGraw-Hill Companies announced it was selling its entire education division to Apollo Global Management for $2.5 billion, on March 22,2013, it announced it had completed the sale and the proceeds were for $2.4 billion in cash. In 2014, McGraw Hill Education India partnered with GreyCampus to promote Online Learning Courses among University Grants Commission- National eligibility Test Aspirants, McGraw Hill Education India is located in Noida area of Delhi/NCR. Operating segments of McGraw-Hill Education include, McGraw-Hill Education K-12, which develops solutions and content for early childhood education, K-12 learners. McGraw-Hill Education Higher Ed, which focuses on post-secondary education, McGraw-Hill Education Professional, focused on post-graduate and professional learners. McGraw-Hill Education International, which focuses on learners and professionals outside of the United States, in 2013, McGraw-Hill Education acquired the entirety of shares in Tata McGraw-Hill Education Private Limited, the companys long-existing joint venture with Tata Group in India
McGraw-Hill
–
McGraw Hill Financial, Inc.
McGraw-Hill
–
1221 Avenue of the Americas, the headquarters of McGraw-Hill
McGraw-Hill
–
2008 conference booth
23.
Formal language
–
In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a language is the set of symbols, letters. The strings formed from this alphabet are called words, and the words belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is defined by means of a formal grammar such as a regular grammar or context-free grammar. The field of language theory studies primarily the purely syntactical aspects of such languages—that is. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift, literally meaning concept writing, axel Thues early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars. The elements of an alphabet are called its letters, alphabets may be infinite, however, most definitions in formal language theory specify finite alphabets, and most results only apply to them. A word over an alphabet can be any sequence of letters. The set of all words over an alphabet Σ is usually denoted by Σ*, the length of a word is the number of letters it is composed of. For any alphabet there is one word of length 0, the empty word. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words, the result of concatenating a word with the empty word is the original word. A formal language L over an alphabet Σ is a subset of Σ*, that is, sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of well-formed expressions. In computer science and mathematics, which do not usually deal with natural languages, in practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the concept of a language. By an abuse of the definition, a formal language is often thought of as being equipped with a formal grammar that describes it. The following rules describe a formal language L over the alphabet Σ =, Every nonempty string that does not contain + or =, a string containing = is in L if and only if there is exactly one =, and it separates two valid strings of L. A string containing + but not = is in L if, no string is in L other than those implied by the previous rules
Formal language
–
Structure of a syntactically well-formed, although nonsensical English sentence (
historical example from Chomsky 1957).
24.
Formal proof
–
A formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system, the notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof, the theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. Formal proofs often are constructed with the help of computers in interactive theorem proving, significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs is usually computationally intractable and/or only semi-decidable, a formal language is a set of finite sequences of symbols. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formal proofs are expressed in some formal language. A formal grammar is a description of the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, an interpretation of a formal system is the assignment of meanings to the symbols, and values to the sentences of a formal system. The study of interpretations is called formal semantics, giving an interpretation is synonymous with constructing a model. Proof Mathematical proof Proof theory Axiomatic system A Special Issue on Formal Proof, notices of the American Mathematical Society. 2πix. com, Logic Part of a series of articles covering mathematics and logic
Formal proof
–
Functional:
25.
Well-formed formula
–
In mathematical logic, a well-formed formula, abbreviated wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language, a formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic, a key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask is φ true, once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, although the term formula may be used for written marks, it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. Thus the same formula may be more than once. They are given meanings by interpretations, for example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula, the formulas of propositional calculus, also called propositional formulas, are expressions such as. Their definition begins with the choice of a set V of propositional variables. The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses, the formulas will be certain expressions over this alphabet. The formulas are inductively defined as follows, Each propositional variable is, on its own, If φ is a formula, then ¬φ is a formula. If φ and ψ are formulas, and • is any binary connective, here • could be the usual operators ∨, ∧, →, or ↔. The sequence of symbols p)) is not a formula, because it does not conform to the grammar, a complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules are assumed among the operators, for example, assuming the precedence 1. Then the formula may be abbreviated as p → q ∧ r → s ∨ ¬q ∧ ¬s This is, however, If the precedence was assumed, for example, to be left-right associative, in following order,1. ∨4. →, then the formula above would be rewritten as → The definition of a formula in first-order logic Q S is relative to the signature of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, the definition of a formula comes in several parts. First, the set of terms is defined recursively, terms, informally, are expressions that represent objects from the domain of discourse
Well-formed formula
–
This diagram shows the
syntactic entities which may be constructed from
formal languages. The
symbols and
strings of symbols may be broadly divided into
nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into
theorems and non-theorems.
26.
Classical logic
–
Classical logic is an intensively studied and widely used class of formal logics. Classical logic was devised as a two-level logical system, with simple semantics for the levels representing true. These judgments find themselves if two pairs of two operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions. George Booles algebraic reformulation of logic, his system of Boolean logic, with the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics, the values are the elements of an arbitrary Boolean algebra, true corresponds to the maximal element of the algebra. Intermediate elements of the algebra correspond to truth values other than true, the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Many-valued logic, including logic, which rejects the law of the excluded middle. Graham Priest, An Introduction to Non-Classical Logic, From If to Is, 2nd Edition, CUP,2008, ISBN 978-0-521-67026-5 Warren Goldfard, Deductive Logic, 1st edition,2003, ISBN 0-87220-660-2
Classical logic
–
Law of excluded middle
27.
Natural deduction
–
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the natural way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the laws of deductive reasoning. Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica. His proposals led to different notations such as Fitch-style calculus or Suppes method of which e. g. Lemmon gave a variant called system L. The term natural deduction was coined in that paper, Ich wollte nun zunächst einmal einen Formalismus aufstellen, so ergab sich ein Kalkül des natürlichen Schließens. Gentzen was motivated by a desire to establish the consistency of number theory and he was unable to prove the main result required for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his system, the sequent calculus. His 1965 monograph Natural deduction, a study was to become a reference work on natural deduction. In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly, the system presented in this article is a minor variation of Gentzens or Prawitzs formulation, but with a closer adherence to Martin-Löfs description of logical judgments and connectives. A judgment is something that is knowable, that is, an object of knowledge and it is evident if one in fact knows it. In mathematical logic however, evidence is not as directly observable. The process of deduction is what constitutes a proof, in other words, the most important judgments in logic are of the form A is true. The letter A stands for any expression representing a proposition, the truth judgments thus require a more primitive judgment, to start with, we shall concern ourselves with the simplest two judgments A is a proposition and A is true, abbreviated as A prop and A true respectively. The judgment A prop defines the structure of proofs of A. For this reason, the rules for this judgment are sometimes known as formation rules. To illustrate, if we have two propositions A and B, then we form the compound proposition A and B, written symbolically as A ∧ B. The general form of a rule is, where each J i is a judgment. The judgments above the line are known as premises, and those below the line are conclusions, other common logical propositions are disjunction, negation, implication, and the logical constants truth and falsehood
Natural deduction
–
Summary of first-order system
28.
Theorem
–
In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
Theorem
–
A
planar map with five colors such that no two regions with the same color meet. It can actually be colored in this way with only four colors. The
four color theorem states that such colorings are possible for any planar map, but every known proof involves a computational search that is too long to check by hand.
29.
Logical consequence
–
Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by the premises, the philosophical analysis of logical consequence involves the questions, In what sense does a conclusion follow from its premises. And What does it mean for a conclusion to be a consequence of premises, All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and this is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using inference rules, for instance, we can express the logical form of a valid argument as, All A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
Logical consequence
–
Tautology
30.
Symbol (formal)
–
A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. In logic, symbols build literal utility to illustrate ideas, symbols of a formal language need not be symbols of anything. For instance there are constants which do not refer to any idea. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them, a symbol or string of symbols may comprise a well-formed formula if it is consistent with the formation rules of the language. In a formal system a symbol may be used as a token in formal operations. The set of symbols in a formal language is referred to as an alphabet A formal symbol as used in first-order logic may be a variable, a constant. Formal symbols are thought of as purely syntactic structures, composed into larger structures using a formal grammar. The move to view units in natural language as formal symbols was initiated by Noam Chomsky, the generative grammar model looked upon syntax as autonomous from semantics. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and this is the philosophical premise underlying Montague grammar. List of mathematical symbols List of logic symbols
Symbol (formal)
–
This diagram shows the
syntactic entities that may be constructed from
formal languages. The symbols and
strings of symbols may be broadly divided into
nonsense and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into
theorems and non-theorems.
31.
Syntax (logic)
–
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the used for constructing, or transforming the symbols and words of a language. Syntax is usually associated with the governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system. In computer science, the term refers to the rules governing the composition of well-formed expressions in a programming language. As in mathematical logic, it is independent of semantics and interpretation, a symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything, for instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language. A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them, a formal language is a syntactic entity which consists of a set of finite strings of symbols which are its words. Which strings of symbols are words is determined by fiat by the creator of the language, usually by specifying a set of formation rules. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formation rules are a precise description of which strings of symbols are the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a proposition is a sentence expressing something true or false. A proposition is identified ontologically as an idea, concept or abstraction whose token instances are patterns of symbols, marks, sounds, propositions are considered to be syntactic entities and also truthbearers. A formal theory is a set of sentences in a formal language, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, Formal systems, like other syntactic entities may be defined without any interpretation given to it. A formula A is a syntactic consequence within some formal system F S of a set Г of formulas if there is a derivation in formal system F S of A from the set Г. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, a formal system S is syntactically complete iff for each formula A of the language of the system either A or ¬A is a theorem of S
Syntax (logic)
–
This diagram shows the syntactic entities which may be constructed from
formal languages. The
symbols and
strings of symbols may be broadly divided into
nonsense and
well-formed formulas. A formal language is identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into
theorems and non-theorems.
32.
Theory (mathematical logic)
–
In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context, an element ϕ ∈ T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom is also a theorem, a first-order theory is a set of first-order sentences. When defining theories for foundational purposes, additional care must be taken, the construction of a theory begins by specifying a definite non-empty conceptual class E, the elements of which are called statements. These initial statements are often called the elements or elementary statements of the theory. A theory T is a class consisting of certain of these elementary statements. The elementary statements which belong to T are called the elementary theorems of T, in this way, a theory is a way of designating a subset of E which consists entirely of true statements. This general way of designating a theory stipulates that the truth of any of its statements is not known without reference to T. Thus the same statement may be true with respect to one theory. This is as in language, where statements such as He is a terrible person. Cannot be judged to be true or false without reference to some interpretation of who He is, a theory S is a subtheory of a theory T if S is a subset of T. If T is a subset of S then S is an extension or supertheory of T A theory is said to be a theory if T is an inductive class. That is, that its content is based on some formal deductive system, in a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory. A syntactically consistent theory is a theory from which not every sentence in the language can be proven. In a deductive system that satisfies the principle of explosion, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory, a satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory, any satisfiable theory is syntactically consistent, because the structure satisfying the theory will satisfy exactly one of φ and the negation of φ, for each sentence φ. A consistent theory is defined to be a syntactically consistent theory. For first-order logic, the most important case, it follows from the theorem that the two meanings coincide
Theory (mathematical logic)
–
Functional:
33.
Term logic
–
This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the terminology and ideas of term logic can have difficulty understanding such texts. Aristotles logical work is collected in the six texts that are known as the Organon. Modern work on Aristotles logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm, the proposition consists of two terms, in which one term is affirmed or denied of the other, and which is capable of truth or falsity. The syllogism is an inference in which one proposition follows of necessity from two others, a proposition may be universal or particular, and it may be affirmative or negative. Aristotles original square of opposition, however, does not lack existential import and this is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid, the theory of the syllogism partly constrains the interpretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the patterns is, Every C is B Every C is A So, some A is B This is invalid if the A form lacks existential import. It is held to be valid, and so we know how the A form is to be interpreted, one then naturally asks about the O form, what do the syllogisms tell us about it. The answer is that they tell us nothing and this is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. But the weakened forms were typically ignored, one other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotles discussion of infinite negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use non for this, we make non-horse, in medieval Latin non and not are the same word, and so the distinction required special discussion. It became common to use infinite negation, and logicians pondered its logic, some writers in the twelfth century and thirteenth centuries adopted a principle called conversion by contraposition. For in the case it leads directly from the truth, Every man is a being to the falsehood. Unfortunately, by Buridans time the principle of contraposition had been advocated by a number of authors, a term is the basic component of the proposition. The original meaning of the horos is extreme or boundary, the two terms lie on the outside of the proposition, joined by the act of affirmation or denial
Term logic
–
Aristotelianism
34.
Syllogism
–
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a statement and a specific statement. For example, knowing that all men are mortal and that Socrates is a man, Syllogistic arguments are usually represented in a three-line form, All men are mortal. In antiquity, two theories of the syllogism existed, Aristotelian syllogistic and Stoic syllogistic. Aristotle defines the syllogism as. a discourse in which certain things having been supposed, despite this very general definition, in Aristotles work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably and this article is concerned only with this traditional use. The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle, the onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle develops his theory of the syllogism. Prior Analytics, upon re-discovery, was regarded by logicians as a closed and complete body of doctrine, leaving very little for thinkers of the day to debate. Aristotles theories on the syllogism for assertoric sentences was considered especially remarkable, Aristotles Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise. Aristotles terminology in this aspect of his theory was deemed vague and in many cases unclear and his original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, boethius contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the twelfth century and his perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelards distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a coherent concept of Aristotles modal syllogism model. For two hundred years after Buridans discussions, little was said about syllogistic logic, the Aristotelian syllogism dominated Western philosophical thought for many centuries. In the 17th century, Sir Francis Bacon rejected the idea of syllogism as being the best way to draw conclusions in nature. Instead, Bacon proposed a more inductive approach to the observation of nature, in the 19th century, modifications to syllogism were incorporated to deal with disjunctive and conditional statements. Kant famously claimed, in Logic, that logic was the one completed science, though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kants opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift. This introduced a calculus, a method of representing categorical statements by the use of quantifiers, in the last 20 years, Bolzanos work has resurfaced and become subject of both translation and contemporary study
Syllogism
–
Relationships between the four types of propositions in the
square of opposition (Black areas are empty, red areas are nonempty.)
35.
Square of opposition
–
The square of opposition is a diagram representing the relations between four categorical propositions. The origin of the square can be traced back to Aristotle making the distinction between two oppositions, contradiction and contrariety, but Aristotle did not draw any diagram. This was done several centuries laters by Apuleius and Boethius, in traditional logic, a proposition is a spoken assertion, not the meaning of an assertion, as in modern philosophy of language and logic. A categorical proposition is a simple proposition containing two terms, subject and predicate, in which the predicate is either asserted or denied of the subject, every categorical proposition can be reduced to one of four logical forms. These are, The so-called A proposition, the universal affirmative, whose form in Latin is omne S est P, the E proposition, the universal negative, Latin form nullum S est P, usually translated as no S are P. The I proposition, the particular affirmative, Latin quoddam S est P, the O proposition, the particular negative, Latin quoddam S non est P, usually translated as some S are not P. Aristotle states, that there are certain logical relationships between four kinds of proposition. He says that to every affirmation there corresponds exactly one negation, and that every affirmation and its negation are opposed such that one of them must be true. A pair of affirmative and negative statements he calls a contradiction, examples of contradictories are every man is white and not every man is white, no man is white and some man is white. Contrary statements, are such that both cannot at the time be true. Examples of these are the universal affirmative every man is white, and these cannot be true at the same time. However, these are not contradictories because both of them may be false, for example, it is false that every man is white, since some men are not white. Yet it is false that no man is white, since there are some white men. Since subcontraries are negations of universal statements, they were called particular statements by the medieval logicians, another logical opposition implied by this, though not mentioned explicitly by Aristotle, is alternation, consisting of subalternation and superalternation. Alternation is a relation between a statement and a universal statement of the same quality such that the particular is implied by the other. The particular is the subaltern of the universal, which is the particulars superaltern, for example, if every man is white is true, its contrary no man is white is false. Therefore the contradictory some man is white is true, similarly the universal no man is white implies the particular not every man is white. In summary, Universal statements are contraries, every man is just and no man is just cannot be together, although one may be true and the other false
Square of opposition
–
Depiction from the 15th century
Square of opposition
–
Square of opposition In the
Venn diagrams, black areas are
empty and red areas are nonempty. The faded arrows and faded red areas apply in traditional logic.
36.
Venn diagram
–
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are then in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
Venn diagram
–
Venn diagram showing which uppercase letter
glyphs are shared by the
Greek,
Latin and
Cyrillic alphabets
37.
Propositional calculus
–
Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post
Propositional calculus
–
Law of excluded middle
38.
Boolean algebra
–
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively. It is thus a formalism for describing logical relations in the way that ordinary algebra describes numeric relations. Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic, according to Huntington, the term Boolean algebra was first suggested by Sheffer in 1913. Boolean algebra has been fundamental in the development of digital electronics and it is also used in set theory and statistics. Booles algebra predated the modern developments in algebra and mathematical logic. In an abstract setting, Boolean algebra was perfected in the late 19th century by Jevons, Schröder, Huntington, in fact, M. H. Stone proved in 1936 that every Boolean algebra is isomorphic to a field of sets. Shannon already had at his disposal the abstract mathematical apparatus, thus he cast his switching algebra as the two-element Boolean algebra, in circuit engineering settings today, there is little need to consider other Boolean algebras, thus switching algebra and Boolean algebra are often used interchangeably. Efficient implementation of Boolean functions is a problem in the design of combinational logic circuits. Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Boolean algebra, thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Boolean algebra is not sufficient to capture logic formulas using quantifiers, the closely related model of computation known as a Boolean circuit relates time complexity to circuit complexity. Whereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth values false and these values are represented with the bits, namely 0 and 1. Addition and multiplication then play the Boolean roles of XOR and AND respectively, Boolean algebra also deals with functions which have their values in the set. A sequence of bits is a commonly used such function, another common example is the subsets of a set E, to a subset F of E is associated the indicator function that takes the value 1 on F and 0 outside F. The most general example is the elements of a Boolean algebra, as with elementary algebra, the purely equational part of the theory may be developed without considering explicit values for the variables. The basic operations of Boolean calculus are as follows, AND, denoted x∧y, satisfies x∧y =1 if x = y =1 and x∧y =0 otherwise. OR, denoted x∨y, satisfies x∨y =0 if x = y =0, NOT, denoted ¬x, satisfies ¬x =0 if x =1 and ¬x =1 if x =0. Alternatively the values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables as follows, the first operation, x → y, or Cxy, is called material implication. If x is then the value of x → y is taken to be that of y
Boolean algebra
–
Figure 2. Venn diagrams for conjunction, disjunction, and complement
39.
Propositional formula
–
In propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a truth value. If the values of all variables in a formula are given. A propositional formula may also be called an expression, a sentence. In some contexts, maintaining the distinction may be of importance, for the purposes of the propositional calculus, propositions are considered to be either simple or compound. Compound propositions are considered to be linked by sentential connectives, some of the most common of which are AND, OR, the linking semicolon, and connective BUT are considered to be expressions of AND. A sequence of sentences are considered to be linked by ANDs. For example, The assertion, This cow is blue and that horse is orange but this horse here is purple. is actually a compound proposition linked by ANDs. Simple propositions are declarative in nature, that is, they make assertions about the condition or nature of an object of sensation e. g. This cow is blue, Theres a coyote, Thus the simple primitive assertions must be about specific objects or specific states of mind. Each must have at least a subject, a verb, probably implies I see a dog but should be rejected as too ambiguous. Example, That purple dog is running, This cow is blue, Switch M31 is closed, This cap is off, for the purposes of the propositional calculus a compound proposition can usually be reworded into a series of simple sentences, although the result will probably sound stilted. Example, This blue pig has wings becomes two sentences in the calculus, This pig has wings AND This pig is blue. In contrast, in the calculus, the first sentence breaks into this pig as the subject. Thus it asserts that object this pig is a member of the class of winged things, the second sentence asserts that object this pig has an attribute blue and thus is a member of the class of blue things. In other words, given a domain of discourse winged things, Thus we have a relationship W between p and, W evaluates to. Likewise for B and p and, B evaluates to, along with the new function symbolism F two new symbols are introduced, ∀, and ∃. Some authors refer to predicate logic with identity to emphasize this extension and these symbols, and well-formed strings of them, are said to represent objects, but in a specific algebraic system these objects do not have meanings. Thus work inside the algebra becomes an exercise in obeying certain laws of the algebras syntax rather than in semantics of the symbols, the meanings are to be found outside the algebra
Propositional formula
–
The engineering symbol for the NAND connective (the 'stroke') can be used to build any propositional formula. The notion that truth (1) and falsity (0) can be defined in terms of this connective is shown in the sequence of NANDs on the left, and the derivations of the four evaluations of a NAND b are shown along the bottom. The more common method is to use the definition of the NAND from the truth table.
40.
Logical connective
–
The most common logical connectives are binary connectives which join two sentences which can be thought of as the functions operands. Also commonly, negation is considered to be a unary connective, logical connectives along with quantifiers are the two main types of logical constants used in formal systems such as propositional logic and predicate logic. Semantics of a logical connective is often, but not always, a logical connective is similar to but not equivalent to a conditional operator. In the grammar of natural languages two sentences may be joined by a grammatical conjunction to form a compound sentence. Some but not all such grammatical conjunctions are truth functions, for example, consider the following sentences, A, Jack went up the hill. B, Jill went up the hill, C, Jack went up the hill and Jill went up the hill. D, Jack went up the hill so Jill went up the hill, the words and and so are grammatical conjunctions joining the sentences and to form the compound sentences and. The and in is a connective, since the truth of is completely determined by and, it would make no sense to affirm. Various English words and word pairs express logical connectives, and some of them are synonymous, examples are, In formal languages, truth functions are represented by unambiguous symbols. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, see well-formed formula for the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives. Logical connectives can be used to more than two statements, so one can speak about n-ary logical connective. For example, the meaning of the statements it is raining, comes from Booles interpretation of logic as an elementary algebra. True, the symbol 1 comes from Booles interpretation of logic as an algebra over the two-element Boolean algebra. False, the symbol 0 comes also from Booles interpretation of logic as a ring, some authors used letters for connectives at some time of the history, u. for conjunction and o. Such a logical connective as converse implication ← is actually the same as material conditional with swapped arguments, thus, in some logical calculi certain essentially different compound statements are logically equivalent. A less trivial example of a redundancy is the equivalence between ¬P ∨ Q and P → Q. There are sixteen Boolean functions associating the input truth values P and Q with four-digit binary outputs and these correspond to possible choices of binary logical connectives for classical logic. Different implementations of classical logic can choose different functionally complete subsets of connectives, One approach is to choose a minimal set, and define other connectives by some logical form, as in the example with the material conditional above
Logical connective
–
Tautology
41.
Truth table
–
In particular, truth tables can be used to show whether a propositional expression is true for all legitimate input values, that is, logically valid. A truth table has one column for each variable. Each row of the table contains one possible configuration of the input variables. See the examples below for further clarification, the Com row indicates whether an operator, op, is commutative - P op Q = Q op P. The L id row shows the operators left identities if it has any - values I such that I op Q = Q, the R id row shows the operators right identities if it has any - values I such that P op I = P. The four combinations of values for p, q, are read by row from the table above. The output function for p, q combination, can be read, by row. Key, The following table is oriented by column, rather than by row, There are four columns rather than four rows, to display the four combinations of p, q, as input. P, T T F Fq, T F T F There are 16 rows in this key, the output row for ↚ is thus 2, F F T F and the 16-row key is Logical operators can also be visualized using Venn diagrams. Logical conjunction is an operation on two values, typically the values of two propositions, that produces a value of true if both of its operands are true. The truth table for p AND q is as follows, In ordinary language terms, for all other assignments of logical values to p and to q the conjunction p ∧ q is false. The truth table for p OR q is as follows, Stated in English, if p, then p ∨ q is p, otherwise p ∨ q is q. The truth table associated with the conditional if p then q. Logical equality is an operation on two values, typically the values of two propositions, that produces a value of true if both operands are false or both operands are true. The truth table for p XNOR q is as follows, So p EQ q is true if p and q have the truth value. Exclusive disjunction is an operation on two values, typically the values of two propositions, that produces a value of true if one but not both of its operands is true. The truth table for p XOR q is as follows, For two propositions, XOR can also be written as ∨. The logical NAND is an operation on two values, typically the values of two propositions, that produces a value of false if both of its operands are true
Truth table
–
Tautology
42.
Predicate logic
–
First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
Predicate logic
–
Law of excluded middle
43.
First-order logic
–
First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
First-order logic
–
1. ∀ x ∃ y L (y, x): Everyone is loved by someone.
44.
Naive set theory
–
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using a formal logic, naive set theory is defined informally and it describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday usage of set theory concepts in contemporary mathematics. Sets are of importance in mathematics, in modern formal treatments. Naive set theory suffices for many purposes, while serving as a stepping-stone towards more formal treatments. A naive theory in the sense of set theory is a non-formalized theory, that is. Then, not, for some, for every are treated as in ordinary mathematics, as a matter of convenience, usage of naive set theory and its formalism prevails even in higher mathematics – including in more formal settings of set theory itself. The first development of set theory was a set theory. It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets, Naive set theory may refer to several very distinct notions. It may refer to Informal presentation of a set theory. Early or later versions of Georg Cantors theory and other informal systems, decidedly inconsistent theories, such as a theory of Gottlob Frege that yielded Russells paradox, and theories of Giuseppe Peano and Richard Dedekind. The assumption that any property may be used to form a set, without restriction, one common example is Russells paradox, there is no set consisting of all sets that do not contain themselves. Thus consistent systems of set theory must include some limitations on the principles which can be used to form sets. Some believe that Georg Cantors set theory was not actually implicated in the set-theoretic paradoxes, one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. Cantors paradox can actually be derived from the above assumption using for P x is a cardinal number, Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when. A naive set theory is not necessarily inconsistent, if it specifies the sets allowed to be considered. This can be done by the means of definitions, which are implicit axioms and it is possible to state all the axioms explicitly, as in the case of Halmos Naive Set Theory, which is actually an informal presentation of the usual axiomatic Zermelo–Fraenkel set theory. It is naive in that the language and notations are those of ordinary informal mathematics, likewise, an axiomatic set theory is not necessarily consistent, i. e. not necessarily free of paradoxes. However, the common systems are generally believed to be consistent
Naive set theory
–
Passage with the original set definition of Georg Cantor
45.
Countable set
–
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A countable set is either a set or a countably infinite set. Some authors use countable set to mean countably infinite alone, to avoid this ambiguity, the term at most countable may be used when finite sets are included and countably infinite, enumerable, or denumerable otherwise. Georg Cantor introduced the term countable set, contrasting sets that are countable with those that are uncountable, today, countable sets form the foundation of a branch of mathematics called discrete mathematics. A set S is countable if there exists a function f from S to the natural numbers N =. If such an f can be found that is also surjective, in other words, a set is countably infinite if it has one-to-one correspondence with the natural number set, N. As noted above, this terminology is not universal, some authors use countable to mean what is here called countably infinite, and do not include finite sets. Alternative formulations of the definition in terms of a function or a surjective function can also be given. In 1874, in his first set theory article, Cantor proved that the set of numbers is uncountable. In 1878, he used one-to-one correspondences to define and compare cardinalities, in 1883, he extended the natural numbers with his infinite ordinals, and used sets of ordinals to produce an infinity of sets having different infinite cardinalities. A set is a collection of elements, and may be described in many ways, one way is simply to list all of its elements, for example, the set consisting of the integers 3,4, and 5 may be denoted. This is only effective for small sets, however, for larger sets, even in this case, however, it is still possible to list all the elements, because the set is finite. Some sets are infinite, these sets have more than n elements for any integer n, for example, the set of natural numbers, denotable by, has infinitely many elements, and we cannot use any normal number to give its size. Nonetheless, it out that infinite sets do have a well-defined notion of size. To understand what this means, we first examine what it does not mean, for example, there are infinitely many odd integers, infinitely many even integers, and infinitely many integers overall. However, it out that the number of even integers. This is because we arrange things such that for every integer, or, more generally, n→2n, see picture. However, not all sets have the same cardinality
Countable set
–
Bijective mapping from integer to even numbers
46.
Domain of a function
–
In mathematics, and more specifically in naive set theory, the domain of definition of a function is the set of input or argument values for which the function is defined. That is, the function provides an output or value for each member of the domain, conversely, the set of values the function takes on as output is termed the image of the function, which is sometimes also referred to as the range of the function. For instance, the domain of cosine is the set of all real numbers, if the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the X-axis. Given a function f, X→Y, the set X is the domain of f, in the expression f, x is the argument and f is the value. One can think of an argument as a member of the domain that is chosen as an input to the function, the image of f is the set of all values assumed by f for all possible x, this is the set. The image of f can be the set as the codomain or it can be a proper subset of it. It is, in general, smaller than the codomain, it is the whole codomain if, a well-defined function must map every element of its domain to an element of its codomain. For example, the function f defined by f =1 / x has no value for f, thus, the set of all real numbers, R, cannot be its domain. In cases like this, the function is defined on R\ or the gap is plugged by explicitly defining f. If we extend the definition of f to f = {1 / x x ≠00 x =0 then f is defined for all real numbers, any function can be restricted to a subset of its domain. The restriction of g, A → B to S, where S ⊆ A, is written g |S, S → B. The natural domain of a function is the set of values for which the function is defined, typically within the reals. For instance the natural domain of square root is the non-negative reals when considered as a real number function, when considering a natural domain, the set of possible values of the function is typically called its range. There are two meanings in current mathematical usage for the notion of the domain of a partial function from X to Y, i. e. a function from a subset X of X to Y. Most mathematicians, including recursion theorists, use the domain of f for the set X of all values x such that f is defined. But some, particularly category theorists, consider the domain to be X, in category theory one deals with morphisms instead of functions. Morphisms are arrows from one object to another, the domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned or at least formulated more abstractly, for example, the notion of restricting a morphism to a subset of its domain must be modified
Domain of a function
–
Illustration showing f, a function from pink domain X to blue and yellow codomain Y. The smaller yellow oval inside Y is the
image of f. Either the image or the codomain also sometimes is called the
range of f.
47.
Codomain
–
In mathematics, the codomain or target set of a function is the set Y into which all of the output of the function is constrained to fall. It is the set Y in the f, X → Y. The codomain is also referred to as the range but that term is ambiguous as it may also refer to the image. The set F is called the graph of the function, the set of all elements of the form f, where x ranges over the elements of the domain X, is called the image of f. In general, the image of a function is a subset of its codomain, thus, it may not coincide with its codomain. Namely, a function that is not surjective has elements y in its codomain for which the equation f = y does not have a solution. An alternative definition of function by Bourbaki, namely as just a functional graph, for example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple. With such a definition functions do not have a codomain, although some still use it informally after introducing a function in the form f, X → Y. For a function f, R → R defined by f, x ↦ x 2, or equivalently f = x 2, the codomain of f is R, but f does not map to any negative number. Thus the image of f is the set R0 +, i. e. the interval [0, an alternative function g is defined thus, g, R → R0 + g, x ↦ x 2. While f and g map a given x to the number, they are not, in this view. A third function h can be defined to demonstrate why, h, x ↦ x, the domain of h must be defined to be R0 +, h, R0 + → R. The compositions are denoted h ∘ f, h ∘ g, on inspection, h ∘ f is not useful. The codomain affects whether a function is a surjection, in that the function is surjective if, in the example, g is a surjection while f is not. The codomain does not affect whether a function is an injection, each matrix represents a map with the domain R2 and codomain R2. Some transformations may have image equal to the codomain but many do not. Take for example the matrix T given by T = which represents a linear transformation that maps the point to, the point is not in the image of T, but is still in the codomain since linear transformations from R2 to R2 are of explicit relevance. Just like all 2×2 matrices, T represents a member of that set, examining the differences between the image and codomain can often be useful for discovering properties of the function in question
Codomain
–
A function f from X to Y. The large blue oval is Y which is the codomain of f. The smaller oval inside Y is the
image of f.
48.
Image (mathematics)
–
In mathematics, an image is the subset of a functions codomain which is the output of the function from a subset of its domain. Evaluating a function at each element of a subset X of the domain, the inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S. Image and inverse image may also be defined for binary relations. The word image is used in three related ways, in these definitions, f, X → Y is a function from the set X to the set Y. If x is a member of X, then f = y is the image of x under f, Y is alternatively known as the output of f for argument x. The image of a subset A ⊆ X under f is the subset f ⊆ Y defined by, f = When there is no risk of confusion and this convention is a common one, the intended meaning must be inferred from the context. This makes the image of f a function whose domain is the set of X. The image f of the entire domain X of f is called simply the image of f, let f be a function from X to Y. The set of all the fibers over the elements of Y is a family of sets indexed by Y, for example, for the function f = x2, the inverse image of would be. Again, if there is no risk of confusion, we may denote f −1 by f −1, the notation f −1 should not be confused with that for inverse function. The notation coincides with the one, though, for bijections. The traditional notations used in the section can be confusing. {\displaystyle f=\left\ The image of the set under f is f =, the image of the function f is. The preimage of a is f −1 =, the preimage of is the empty set. F, R → R defined by f = x2, the image of under f is f =, and the image of f is R+. The preimage of f is f −1 =. The preimage of set N = under f is the empty set, F, R2 → R defined by f = x2 + y2. The fibres f −1 are concentric circles about the origin, the origin itself, and the empty set, depending on whether a >0, a =0, or a <0, respectively
Image (mathematics)
–
f is a function from domain X to codomain Y. The smaller oval inside Y is the image of f.
49.
Map (mathematics)
–
There are also a few, less common uses in logic and graph theory. In many branches of mathematics, the map is used to mean a function. For instance, a map is a function in topology. Some authors, such as Serge Lang, use only to refer to maps in which the codomain is a set of numbers. Sets of maps of special kinds are the subjects of many important theories, see for instance Lie group, mapping class group, in the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, all these usages can be applied to maps as general functions or as functions with special properties. In category theory, map is used as a synonym for morphism or arrow. In formal logic, the map is sometimes used for a functional predicate. In graph theory, a map is a drawing of a graph on a surface without overlapping edges, if the surface is a plane then a map is a planar graph, similar to a political map
Map (mathematics)
–
An example of a map in
graph theory.
50.
Function (mathematics)
–
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
Function (mathematics)
–
A function f takes an input x, and returns a single output f (x). One metaphor describes the function as a "machine" or "
black box " that for each input returns a corresponding output.