Mathematics is the study of topics such as quantity, structure 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, mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, 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 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, medicine and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians 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 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
In mathematics, an equivalence relation is a binary relation that is at the same time a reflexive relation, a symmetric relation and a transitive relation. As a consequence of these properties an equivalence relation provides a partition of a set into equivalence classes, a given binary relation ~ on a set X is said to be an equivalence relation if and only if it is reflexive and transitive. That is, for all a, b and c in X, a ~ b if and only if b ~ a. if a ~ b and b ~ c a ~ c. X together with the relation ~ is called a setoid, the equivalence class of a under ~, denoted, is defined as =. Let the set have the equivalence relation, the following sets are equivalence classes of this relation, =, = =. The set of all classes for this relation is. The following are all equivalence relations, Has the same birthday as on the set of all people, is similar to on the set of all triangles. Is congruent to on the set of all triangles, is congruent to, modulo n on the integers. Has the same image under a function on the elements of the domain of the function, has the same absolute value on the set of real numbers Has the same cosine on the set of all angles.
The relation ≥ between real numbers is reflexive and transitive, but not symmetric, for example,7 ≥5 does not imply that 5 ≥7. It is, however, a total order, the relation has a common factor greater than 1 with between natural numbers greater than 1, is reflexive and symmetric, but not transitive. The empty relation R on a non-empty set X is vacuously symmetric and transitive, a partial order is a relation that is reflexive and transitive. Equality is both a relation and a partial order. Equality is the relation on a set that is reflexive. In algebraic expressions, equal variables may be substituted for one another, the equivalence classes of an equivalence relation can substitute for one another, but not individuals within a class. A strict partial order is irreflexive and asymmetric, a partial equivalence relation is transitive and symmetric. Transitive and symmetric imply reflexive if and only if for all a ∈ X, a reflexive and symmetric relation is a dependency relation, if finite, and a tolerance relation if infinite. A preorder is reflexive and transitive, a congruence relation is an equivalence relation whose domain X is the underlying set for an algebraic structure, and which respects the additional structure
Intersection (set theory)
In mathematics, the intersection A ∩ B of two sets A and B is the set that contains all elements of A that belong to B, but no other elements. For explanation of the used in this article, refer to the table of mathematical symbols. The intersection of A and B is written A ∩ B, formally, A ∩ B = that is x ∈ A ∩ B if and only if x ∈ A and x ∈ B. For example, The intersection of the sets and is, the number 9 is not in the intersection of the set of prime numbers and the set of odd numbers. More generally, one can take the intersection of sets at once. The intersection of A, B, C, and D, Intersection is an associative operation, thus, A ∩ = ∩ C. Additionally, intersection is commutative, thus A ∩ B = B ∩ A, inside a universe U one may define the complement Ac of A to be the set of all elements of U not in A. We say that A intersects B if A intersects B at some element, a intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B, in plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A ∩ B = ∅, for example, the sets and are disjoint, the set of even numbers intersects the set of multiples of 3 at 0,6,12,18 and other numbers.
The most general notion is the intersection of a nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, x is an element of the intersection of M if, the notation for this last concept can vary considerably. Set theorists will sometimes write ⋂M, while others will instead write ⋂A∈M A, the latter notation can be generalized to ⋂i∈I Ai, which refers to the intersection of the collection. Here I is a nonempty set, and Ai is a set for every i in I. In the case that the index set I is the set of numbers, notation analogous to that of an infinite series may be seen. When formatting is difficult, this can be written A1 ∩ A2 ∩ A3 ∩, even though strictly speaking, A1 ∩ (A2 ∩ (A3 ∩. Finally, let us note that whenever the symbol ∩ is placed before other symbols instead of them, it should be of a larger size. Note that in the section we excluded the case where M was the empty set
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 and blue. The relation is an element of, 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, Set Theory, Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY, Dover Publications, Inc
Union (set theory)
In set theory, the union of a collection of sets is the set of all elements in the collection. It is one of the operations through which sets can be combined and related to each other. For explanation of the used in this article, refer to the table of mathematical symbols. The union of two sets A and B is the set of elements which are in A, in B, for example, if A = and B = A ∪ B =. Sets cannot have duplicate elements, so the union of the sets and is, multiple occurrences of identical elements have no effect on the cardinality of a set or its contents. Binary union is an operation, that is, A ∪ = ∪ C. The operations can be performed in any order, and the parentheses may be omitted without ambiguity, union is commutative, so the sets can be written in any order. The empty set is an identity element for the operation of union and that is, A ∪ ∅ = A, for any set A. This follows from analogous facts about logical disjunction, since sets with unions and intersections form a Boolean algebra, intersection distributes over union A ∩ = ∪ and union distributes over intersection A ∪ = ∩.
One can take the union of several sets simultaneously, for example, the union of three sets A, B, and C contains all elements of A, all elements of B, and all elements of C, and nothing else. Thus, x is an element of A ∪ B ∪ C if and only if x is in at least one of A, B, and C. In mathematics a finite union means any union carried out on a number of sets. The most general notion is the union of a collection of sets. If M is a set whose elements are themselves sets, x is an element of the union of M if, in symbols, x ∈ ⋃ M ⟺ ∃ A ∈ M, x ∈ A. This idea subsumes the preceding sections, in that A ∪ B ∪ C is the union of the collection, also, if M is the empty collection, the union of M is the empty set. The notation for the concept can vary considerably. For a finite union of sets S1, S2, S3, …, S n one often writes S1 ∪ S2 ∪ S3 ∪ ⋯ ∪ S n or ⋃ i =1 n S i. In the case that the index set I is the set of natural numbers, whenever the symbol ∪ is placed before other symbols instead of between them, it is of a larger size
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 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, 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
Hyperplane separation theorem
In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions, in another version, if both disjoint convex sets are open, there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, the hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces, a related result is the supporting hyperplane theorem. In geometry, a maximum-margin hyperplane is a hyperplane which separates two clouds of points and is at distance from the two. The margin between the hyperplane and the clouds is maximal, see the article on Support Vector Machines for more details. The proof is based on the following lemma, Proof of lemma, Let x j be a sequence in K such that | x j | → δ. Note that /2 is in K since K is convex, ◻ Proof of theorem, Given disjoint nonempty convex sets A, B, let K = A + =.
Since − B is convex and the sum of convex sets is convex, by the lemma, the closure K ¯ of K, which is convex, contains a vector v of minimum norm. Hence, for any x in A and y in B, we have, thus, if v is nonzero, the proof is complete since inf x ∈ A ⟨ x, v ⟩ ≥ | v |2 + sup y ∈ B ⟨ y, v ⟩. More generally, let us first take the case when the interior of K is nonempty, the interior can be exhausted by nonempty compact convex subsets K n, n =1,2, …. Since 0 is not in K, each K n contains a vector v n of minimum length and by the argument in the early part, we have. We can normalize the v n s to have length one, the sequence v n contains a convergent subsequence with limit v, which is nonzero. We have ⟨ x, v ⟩ ≥0 for any x in the interior of K and we now finish the proof as before. Finally, if K has empty interior, the set that it spans has dimension less than that of the whole space. Consequently K is contained in some hyperplane ⟨ ⋅, v ⟩ = c, thus, ⟨ x, v ⟩ ≥ c for all x in K, ◻ The number of dimensions must be finite.
In infinite-dimensional spaces there are examples of two closed, disjoint sets which cannot be separated by a closed hyperplane even in the sense where the inequalities are not strict. The above proof proves the first version of the mentioned in the lede Here
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds.
It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti.
Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
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, 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 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
Introduction to Algorithms
Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. It is used as the textbook for algorithms courses at universities and is commonly cited as a reference for algorithms in published papers. The book sold half a million copies during its first 20 years and its fame has led to the common use of the abbreviation CLRS, or, in the first edition, CLR. The first edition of the textbook did not include Stein as an author and it included 2 chapters that were dropped in the second edition. After the addition of the author in the second edition. This first edition of the book was known as The Big White Book. With the second edition, the predominant color of the changed to green. A third edition was published in August 2009, the mobile depicted on the cover, Big Red by Alexander Calder, can be found at the Whitney Museum of American Art in New York City. C Counting and Probability D Matrices Cormen, Thomas H. Leiserson, Charles E. Rivest, Thomas H.
Leiserson, Charles E. Rivest, Ronald L. Stein, Clifford. Cormen, Thomas H. Leiserson, Charles E. Rivest, Ronald L. Stein, the Art of Computer Programming Official websites by MIT Press MIT lecture MIT6. 046J /18. 410J Introduction to Algorithms - Fall 2005. Held in part by coauthor Charles Leiserson, released as part of MIT OpenCourseWare. Video recordings and transcripts of the lectures, includes slides automatically synchronized to video content. Exercise Solutions While there are no solutions, the following may be helpful