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Mathematics
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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
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Real number
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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
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Ordered field
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In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of a field was abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder. This grew eventually into the Artin–Schreier theory of ordered fields and formally real fields, an ordered field necessarily has characteristic 0 since the elements 0 <1 <1 +1 <1 +1 +1 <. Thus, an ordered field contains an infinite number of elements. Every subfield of a field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is isomorphic to the rational numbers, any Dedekind-complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered field and this implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1. Every ordered field is a real field. There are two equivalent common definitions of an ordered field, the definition of total order appeared first historically and is a first-order axiomatization of the ordering ≤ as a binary predicate. Artin and Schreier gave the definition in terms of positive cone in 1926, although the latter is higher-order, viewing positive cones as maximal prepositive cones provides a larger context in which field orderings are extremal partial orderings. The symbol for multiplication will be henceforth omitted, a prepositive cone or preordering of a field F is a subset P ⊂ F that has the following properties, For x and y in P, both x + y and xy are in P. If x is in F, then x2 is in P, the element −1 is not in P. A preordered field is an equipped with a preordering P. Its non-zero elements P∗ form a subgroup of the group of F. If in addition, the set F is the union of P and −P, the non-zero elements of P are called the positive elements of F. An ordered field is a field F together with a positive cone P, the preorderings on F are precisely the intersections of families of positive cones on F. The positive cones are the maximal preorderings, there is a bijection between the field orderings of F and the positive cones of F. This total ordering ≤P satisfies the properties of the first definition and this ordered field is not Archimedean
4.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Formally, a field is a set together with two operations the addition and the multiplication, which have the properties, called axioms of fields. An operation is a mapping that associates an element of the set to every pair of its elements, the result of the addition of a and b is called the sum of a and b and denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, associativity of addition and multiplication For all a, b and c in F, one has a + = + c and a · = · c. Commutativity of addition and multiplication For all a and b in F one has a + b = b + a and a · b = b · a. Existence of additive and multiplicative identity elements There exists an element 0 in F, called the identity, such that for all a in F. There is an element 1, different from 0 and called the identity, such that for all a in F. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element in F, denoted −a, such that a + =0. For every a ≠0 in F, there exists an element in F, denoted a−1, 1/a, or 1/a, distributivity of multiplication over addition For all a, b and c in F, one has a · = +. The elements 0 and 1 being required to be distinct, a field has, at least, for every a in F, one has − a = ⋅ a. Thus, the inverse of every element is known as soon as one knows the additive inverse of 1. A subtraction and a division are defined in every field by a − b = a +, a subfield E of a field F is a subset of F that contains 1, and is closed under addition, multiplication, additive inverse and multiplicative inverse of a nonzero element. It is straightforward to verify that a subfield is indeed a field, two groups are associated to every field. The field itself is a group under addition, when considering this group structure rather the field structure, one talks of the additive group of the field
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Isomorphism
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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
6.
Rational number
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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 decimal expansion of a rational number always either terminates after a finite 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. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, 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. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0
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Associative property
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In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a rule of replacement for expressions in logical proofs. That is, rearranging the parentheses in such an expression will not change its value, consider the following equations, +4 =2 + =92 × = ×4 =24. Even though the parentheses were rearranged on each line, the values of the expressions were not altered, since this holds true when performing addition and multiplication on any real numbers, it can be said that addition and multiplication of real numbers are associative operations. Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result. For example, the order doesnt matter in the multiplication of numbers, that is. Associative operations are abundant in mathematics, in fact, many algebraic structures explicitly require their binary operations to be associative, however, many important and interesting operations are non-associative, some examples include subtraction, exponentiation and the vector cross product. Z = x = xyz for all x, y, z in S, the associative law can also be expressed in functional notation thus, f = f. If a binary operation is associative, repeated application of the produces the same result regardless how valid pairs of parenthesis are inserted in the expression. This is called the generalized associative law, thus the product can be written unambiguously as abcd. As the number of elements increases, the number of ways to insert parentheses grows quickly. Some examples of associative operations include the following, the two methods produce the same result, string concatenation is associative. In arithmetic, addition and multiplication of numbers are associative, i. e. + z = x + = x + y + z z = x = x y z } for all x, y, z ∈ R. x, y, z\in \mathbb. }Because of associativity. Addition and multiplication of numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative, the greatest common divisor and least common multiple functions act associatively. Gcd = gcd = gcd lcm = lcm = lcm } for all x, y, z ∈ Z. x, y, z\in \mathbb. }Taking the intersection or the union of sets, ∩ C = A ∩ = A ∩ B ∩ C ∪ C = A ∪ = A ∪ B ∪ C } for all sets A, B, C. Slightly more generally, given four sets M, N, P and Q, with h, M to N, g, N to P, in short, composition of maps is always associative. Consider a set with three elements, A, B, and C, thus, for example, A=C = A
8.
Commutative property
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says 3 +4 =4 +3 or 2 ×5 =5 ×2, the property can also be used in more advanced settings. The name is needed there are operations, such as division and subtraction. The commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a binary operation then the two elements are said to commute under that operation. The term commutative is used in several related senses, putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same, in contrast, putting on underwear and trousers is not commutative. The commutativity of addition is observed when paying for an item with cash, regardless of the order the bills are handed over in, they always give the same total. The multiplication of numbers is commutative, since y z = z y for all y, z ∈ R For example,3 ×5 =5 ×3. Some binary truth functions are also commutative, since the tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p and this function is also written as p IFF q, or as p ≡ q, or as Epq. Further examples of binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors. Concatenation, the act of joining character strings together, is a noncommutative operation, rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. The twists of the Rubiks Cube are noncommutative and this can be studied using group theory. Some non-commutative binary operations, Records of the use of the commutative property go back to ancient times. The Egyptians used the property of multiplication to simplify computing products. Euclid is known to have assumed the property of multiplication in his book Elements
9.
Reflexive relation
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In mathematics, a binary relation R over a set X is reflexive if every element of X is related to itself. In mathematical notation, this is, ∀ a ∈ X An example of a relation is the relation is equal to on the set of real numbers. A reflexive relation is said to have the property or is said to possess reflexivity. A relation that is irreflexive, or anti-reflexive, is a relation on a set where no element is related to itself. An example is the greater than relation on the real numbers, note that not every relation which is not reflexive is irreflexive, it is possible to define relations where some elements are related to themselves but others are not. A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is related to itself, formally, if ∀x, y∈S. The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~, equivalently, it is the union of ~ and the identity relation on S, formally, = ∪. For example, the closure of x<y is x≤y. The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆ shares the same reflexive closure as ~ and it can be seen in a way as the opposite of the reflexive closure. It is equivalent to the complement of the identity relation on S with regard to ~, formally and that is, it is equivalent to ~ except for where x~x is true. For example, the reduction of x≤y is x<y. Authors in philosophical logic often use deviating designations, a reflexive and a quasi-reflexive relation in the mathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively. Binary relation Symmetric relation Antisymmetric relation Transitive relation Levy, A, basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. ISBN 0-486-42079-5 Lidl, R. and Pilz, G, applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag. Hazewinkel, Michiel, ed. Reflexivity, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
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Antisymmetric relation
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In mathematics, a binary relation R on a set X is anti-symmetric if there is no pair of distinct elements of X each of which is related by R to the other. More formally, R is anti-symmetric precisely if for all a and b in X if R and R, then a = b, or, equivalently, if R with a ≠ b, then R must not hold. As a simple example, the divisibility order on the numbers is an anti-symmetric relation. In mathematical notation, this is, ∀ a, b ∈ X, ⇒ a = b or, equivalently, ∀ a, b ∈ X. The usual order relation ≤ on the numbers is anti-symmetric. A relation can be symmetric and anti-symmetric, and there are relations which are neither symmetric nor anti-symmetric. Anti-symmetry is different from asymmetry, which requires both anti-symmetry and irreflexivity, the relation x is even, y is odd between a pair of integers is anti-symmetric, Every asymmetric relation is also an anti-symmetric relation. Symmetric relation Asymmetric relation Symmetry in mathematics Weisstein, Eric W. Antisymmetric Relation, theory and Problems of Discrete Mathematics
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Upper and lower bounds
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In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of K which is greater than or equal to every element of S. The term lower bound is defined dually as an element of K which is less than or equal to every element of S. A set with an upper bound is said to be bounded from above by that bound, the terms bounded above are also used in the mathematical literature for sets that have upper bounds. For example,5 is a bound for the set, so is 4. Another example, for the set, the number 42 is both an upper bound and a bound, all other real numbers are either an upper bound or a lower bound for that set. Every subset of the numbers has a lower bound, since the natural numbers have a least element. An infinite subset of the numbers cannot be bounded from above. An infinite subset of the integers may be bounded from below or bounded from above, an infinite subset of the rational numbers may or may not be bounded from below and may or may not be bounded from above. Every finite subset of a non-empty totally ordered set has both upper and lower bounds, the definitions can be generalized to functions and even sets of functions. Given a function f with domain D and an ordered set as codomain. The upper bound is called sharp if equality holds for at least one value of x, function g defined on domain D and having the same codomain is an upper bound of f if g ≥ f for each x in D. Function g is said to be an upper bound of a set of functions if it is an upper bound of each function in that set. The notion of lower bound for functions is defined analogously, with ≤ replacing ≥, an upper bound is said to be a tight upper bound, a least upper bound, or a supremum if no smaller value is an upper bound. Similarly a lower bound is said to be a lower bound
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Infimum and supremum
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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound is also commonly used, the supremum of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists. Consequently, the supremum is also referred to as the least upper bound, the infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are special cases that are important in analysis. However, the general definitions remain valid in the abstract setting of order theory where arbitrary partially ordered sets are considered. The concepts of infimum and supremum are similar to minimum and maximum, for instance, the positive real numbers ℝ+* does not have a minimum, because any given element of ℝ+* could simply be divided in half resulting in a smaller number that is still in ℝ+*. There is, however, exactly one infimum of the real numbers,0. A lower bound of a subset S of an ordered set is an element a of P such that a ≤ x for all x in S. A lower bound a of S is called an infimum of S if for all lower bounds y of S in P, y ≤ a. Similarly, a bound of a subset S of a partially ordered set is an element b of P such that b ≥ x for all x in S. An upper bound b of S is called a supremum of S if for all upper bounds z of S in P, z ≥ b, infima and suprema do not necessarily exist. Existence of an infimum of a subset S of P can fail if S has no lower bound at all, however, if an infimum or supremum does exist, it is unique. Consequently, partially ordered sets for which certain infima are known to exist become especially interesting, more information on the various classes of partially ordered sets that arise from such considerations are found in the article on completeness properties. If the supremum of a subset S exists, it is unique, if S contains a greatest element, then that element is the supremum, otherwise, the supremum does not belong to S. Likewise, if the infimum exists, it is unique. If S contains a least element, then that element is the infimum, otherwise, the infimum of a subset S of a partially ordered set P, assuming it exists, does not necessarily belong to S. If it does, it is a minimal or least element of S. Similarly, if the supremum of S belongs to S, for example, consider the set of negative real numbers. This set has no greatest element, since for every element of the set, there is another, larger, for instance, for any negative real number x, there is another negative real number x 2, which is greater. On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set, hence,0 is the least upper bound of the negative reals, so the supremum is 0
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Least-upper-bound property
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In mathematics, the least-upper-bound property is a fundamental property of the real numbers and certain other ordered sets. A set X has the property if and only if every non-empty subset of X with an upper bound has a least upper bound in X. The least-upper-bound property is one form of the axiom for the real numbers. It is usually taken as an axiom in synthetic constructions of the real numbers, in order theory, this property can be generalized to a notion of completeness for any partially ordered set. A linearly ordered set that is dense and has the least upper bound property is called a linear continuum, let S be a non-empty set of real numbers. A real number x is called a bound for S if x ≥ s for all s ∈ S. A real number x is the least upper bound for S if x is a bound for S and x ≤ y for every upper bound y of S. The least-upper-bound property states that any non-empty set of numbers that has an upper bound must have a least upper bound in real numbers. More generally, one may define upper bound and least upper bound for any subset of an ordered set X. In this case, we say that X has the property if every non-empty subset of X with an upper bound has a least upper bound. For example, the set Q of rational numbers does not have the property under the usual order. For instance, the set = Q ∩ has a bound in Q. The construction of the numbers using Dedekind cuts takes advantage of this failure by defining the irrational numbers as the least upper bounds of certain subsets of the rationals. The least-upper-bound property is equivalent to forms of the completeness axiom. It is possible to prove the least-upper-bound property using the assumption that every Cauchy sequence of real numbers converges, let S be a nonempty set of real numbers, and suppose that S has an upper bound B1. Since S is nonempty, there exists a real number A1 that is not a bound for S. Define sequences A1, A2, A3. and B1, B2. Recursively as follows, Check whether ⁄2 is a bound for S. If it is, let An+1 = An and let Bn+1 = ⁄2, otherwise there must be an element s in S so that s> ⁄2
14.
Archimedean property
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Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. An algebraic structure in any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, an ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with different formulations. The concept was named by Otto Stolz after the ancient Greek geometer, the Archimedean property appears in Book V of Euclids Elements as Definition 4, Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is known as the Theorem of Eudoxus or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs, Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y if, for natural number n, the multiple nx is less than y, that is. The group G is Archimedean if there is no x, y such that x is infinitesimal with respect to y. Additionally, if K is a structure with a unit — for example. If x is infinitesimal with respect to 1, then x is an infinitesimal element, likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements, an ordered field has some additional properties. One may assume that the numbers are contained in the field. If x is infinitesimal, then 1/x is infinite, and vice versa, therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a number, then r x is also infinitesimal. As a result, given an element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal
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Bijection
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In mathematical terms, a bijective function f, X → Y is a one-to-one and onto mapping of a set X to a set Y. A bijection from the set X to the set Y has a function from Y to X. If X and Y are finite sets, then the existence of a means they have the same number of elements. For infinite sets the picture is complicated, leading to the concept of cardinal number. A bijective function from a set to itself is called a permutation. Bijective functions are essential to many areas of including the definitions of isomorphism, homeomorphism, diffeomorphism, permutation group. Satisfying properties and means that a bijection is a function with domain X and it is more common to see properties and written as a single statement, Every element of X is paired with exactly one element of Y. Functions which satisfy property are said to be onto Y and are called surjections, Functions which satisfy property are said to be one-to-one functions and are called injections. With this terminology, a bijection is a function which is both a surjection and an injection, or using words, a bijection is a function which is both one-to-one and onto. Consider the batting line-up of a baseball or cricket team, the set X will be the players on the team and the set Y will be the positions in the batting order The pairing is given by which player is in what position in this order. Property is satisfied since each player is somewhere in the list, property is satisfied since no player bats in two positions in the order. Property says that for each position in the order, there is some player batting in that position, in a classroom there are a certain number of seats. A bunch of students enter the room and the instructor asks them all to be seated. After a quick look around the room, the instructor declares that there is a bijection between the set of students and the set of seats, where each student is paired with the seat they are sitting in. The instructor was able to conclude there were just as many seats as there were students. For any set X, the identity function 1X, X → X, the function f, R → R, f = 2x +1 is bijective, since for each y there is a unique x = /2 such that f = y. In more generality, any linear function over the reals, f, R → R, f = ax + b is a bijection, each real number y is obtained from the real number x = /a. The function f, R →, given by f = arctan is bijective since each real x is paired with exactly one angle y in the interval so that tan = x
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Injective
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In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness, it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the codomain is the image of at most one element of its domain. The term one-to-one function must not be confused with one-to-one correspondence, occasionally, an injective function from X to Y is denoted f, X ↣ Y, using an arrow with a barbed tail. A function f that is not injective is sometimes called many-to-one, however, the injective terminology is also sometimes used to mean single-valued, i. e. each argument is mapped to at most one value. A monomorphism is a generalization of a function in category theory. Let f be a function whose domain is a set X, the function f is said to be injective provided that for all a and b in X, whenever f = f, then a = b, that is, f = f implies a = b. Equivalently, if a ≠ b, then f ≠ f, in particular the identity function X → X is always injective. If the domain X = ∅ or X has only one element, the function f, R → R defined by f = 2x +1 is injective. The function g, R → R defined by g = x2 is not injective, however, if g is redefined so that its domain is the non-negative real numbers [0, +∞), then g is injective. The exponential function exp, R → R defined by exp = ex is injective, the natural logarithm function ln, → R defined by x ↦ ln x is injective. The function g, R → R defined by g = xn − x is not injective, since, for example, g = g =0. More generally, when X and Y are both the real line R, then a function f, R → R is one whose graph is never intersected by any horizontal line more than once. This principle is referred to as the line test. Functions with left inverses are always injections and that is, given f, X → Y, if there is a function g, Y → X such that, for every x ∈ X g = x then f is injective. In this case, g is called a retraction of f, conversely, f is called a section of g. Conversely, every injection f with non-empty domain has an inverse g. Note that g may not be an inverse of f because the composition in the other order, f o g. In other words, a function that can be undone or reversed, injections are reversible but not always invertible
17.
Surjective function
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It is not required that x is unique, the function f may map one or more elements of X to the same element of Y. The French prefix sur means over or above and relates to the fact that the image of the domain of a surjective function completely covers the functions codomain, any function induces a surjection by restricting its codomain to its range. Every surjective function has an inverse, and every function with a right inverse is necessarily a surjection. The composite of surjective functions is always surjective, any function can be decomposed into a surjection and an injection. A surjective function is a function whose image is equal to its codomain, equivalently, a function f with domain X and codomain Y is surjective if for every y in Y there exists at least one x in X with f = y. Surjections are sometimes denoted by a two-headed rightwards arrow, as in f, X ↠ Y, symbolically, If f, X → Y, then f is said to be surjective if ∀ y ∈ Y, ∃ x ∈ X, f = y. For any set X, the identity function idX on X is surjective, the function f, Z → defined by f = n mod 2 is surjective. The function f, R → R defined by f = 2x +1 is surjective, because for every real number y we have an x such that f = y, an appropriate x is /2. However, this function is not injective since e. g. the pre-image of y =2 is, the function g, R → R defined by g = x2 is not surjective, because there is no real number x such that x2 = −1. However, the g, R → R0+ defined by g = x2 is surjective because for every y in the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y. The natural logarithm ln, → R is a surjective. Its inverse, the function, is not surjective as its range is the set of positive real numbers. The matrix exponential is not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as a map from the space of all n×n matrices to the linear group of degree n, i. e. the group of all n×n invertible matrices. Under this definition the matrix exponential is surjective for complex matrices, the projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty. In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function, a function is bijective if and only if it is both surjective and injective. If a function is identified with its graph, then surjectivity is not a property of the function itself, unlike injectivity, surjectivity cannot be read off of the graph of the function alone. The function g, Y → X is said to be an inverse of the function f, X → Y if f = y for every y in Y
18.
Alfred Tarski
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Alfred Tarski was a renowned Polish logician, mathematician and philosopher. Tarski taught and carried out research in mathematics at the University of California, Alfred Tarski was born Alfred Teitelbaum, to parents who were Polish Jews in comfortable circumstances relative to other Jews in the overall region. He first manifested his mathematical abilities while in school, at Warsaws Szkoła Mazowiecka. Nevertheless, he entered the University of Warsaw in 1918 intending to study biology, Leśniewski recognized Tarskis potential as a mathematician and encouraged him to abandon biology. Tarski and Leśniewski soon grew cool to each other, however, in later life, Tarski reserved his warmest praise for Kotarbiński, as was mutual. In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to Tarski, the Tarski brothers also converted to Roman Catholicism, Polands dominant religion. Alfred did so even though he was an avowed atheist, Tarski was a Polish nationalist who saw himself as a Pole and wished to be fully accepted as such — later, in America, he spoke Polish at home. In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background and she had worked as a courier for the army in the Polish–Soviet War. They had two children, a son Jan who became a physicist, and a daughter Ina who married the mathematician Andrzej Ehrenfeucht, Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russells recommendation it was awarded to Leon Chwistek. In 1930, Tarski visited the University of Vienna, lectured to Karl Mengers colloquium, thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Mengers research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, in 1937, Tarski applied for a chair at Poznań University but the chair was abolished. Tarskis ties to the Unity of Science movement likely saved his life, thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities, in 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Tarski became an American citizen in 1945, although emeritus from 1968, he taught until 1973 and supervised Ph. D. candidates until his death. At Berkeley, Tarski acquired a reputation as an awesome and demanding teacher, Tarski was extroverted, quick-witted, strong-willed, energetic, and sharp-tongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority, some students were frightened away, but a circle of disciples remained, many of whom became world-renowned leaders in the field
19.
Set (mathematics)
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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
20.
Abelian group
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That is, these are the groups that obey the axiom of commutativity. Abelian groups generalize the arithmetic of addition of integers and they are named after Niels Henrik Abel. The concept of a group is one of the first concepts encountered in undergraduate abstract algebra, from which many other basic concepts, such as modules. The theory of groups is generally simpler than that of their non-abelian counterparts. On the other hand, the theory of abelian groups is an area of current research. An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b, the symbol • is a general placeholder for a concretely given operation. Identity element There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = a holds. Inverse element For each a in A, there exists an element b in A such that a • b = b • a = e, commutativity For all a, b in A, a • b = b • a. A group in which the operation is not commutative is called a non-abelian group or non-commutative group. There are two main conventions for abelian groups – additive and multiplicative. Generally, the notation is the usual notation for groups, while the additive notation is the usual notation for modules. To verify that a group is abelian, a table – known as a Cayley table – can be constructed in a similar fashion to a multiplication table. If the group is G = under the operation ⋅, the th entry of this contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal and this is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the th entry of the table equals the th entry, every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form a group under addition, as do the integers modulo n. Every ring is a group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group, in particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication
21.
Georg Cantor
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Georg Ferdinand Ludwig Philipp Cantor was a German mathematician. He invented set theory, which has become a theory in mathematics. In fact, Cantors method of proof of this theorem implies the existence of an infinity of infinities and he defined the cardinal and ordinal numbers and their arithmetic. Cantors work is of great philosophical interest, a fact of which he was well aware, E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Cantor, a devout Lutheran, believed the theory had been communicated to him by God, Kronecker objected to Cantors proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. The harsh criticism has been matched by later accolades, in 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics. David Hilbert defended it from its critics by declaring, From his paradise that Cantor with us unfolded, we hold our breath in awe, knowing, we shall not be expelled. Georg Cantor was born in the merchant colony in Saint Petersburg, Russia. Georg, the oldest of six children, was regarded as an outstanding violinist and his grandfather Franz Böhm was a well-known musician and soloist in a Russian imperial orchestra. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt, his skills in mathematics. In 1862, Cantor entered the Swiss Federal Polytechnic and he spent the summer of 1866 at the University of Göttingen, then and later a center for mathematical research. Cantor submitted his dissertation on number theory at the University of Berlin in 1867, after teaching briefly in a Berlin girls school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the habilitation for his thesis, also on number theory. In 1874, Cantor married Vally Guttmann and they had six children, the last born in 1886. Cantor was able to support a family despite modest academic pay, during his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday. Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879, however, his work encountered too much opposition for that to be possible. Worse yet, Kronecker, a figure within the mathematical community and Cantors former professor. Cantor came to believe that Kroneckers stance would make it impossible for him ever to leave Halle, in 1881, Cantors Halle colleague Eduard Heine died, creating a vacant chair
22.
Richard Dedekind
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Julius Wilhelm Richard Dedekind was a German mathematician who made important contributions to abstract algebra, algebraic number theory and the definition of the real numbers. Dedekinds father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig, as an adult, he never used the names Julius Wilhelm. He was born, lived most of his life, and died in Braunschweig and he first attended the Collegium Carolinum in 1848 before transferring to the University of Göttingen in 1850. There, Dedekind was taught number theory by professor Moritz Stern, Gauss was still teaching, although mostly at an elementary level, and Dedekind became his last student. Dedekind received his doctorate in 1852, for a thesis titled Über die Theorie der Eulerschen Integrale and this thesis did not display the talent evident by Dedekinds subsequent publications. At that time, the University of Berlin, not Göttingen, was the facility for mathematical research in Germany. Thus Dedekind went to Berlin for two years of study, where he and Bernhard Riemann were contemporaries, they were awarded the habilitation in 1854. Dedekind returned to Göttingen to teach as a Privatdozent, giving courses on probability and he studied for a while with Peter Gustav Lejeune Dirichlet, and they became good friends. Because of lingering weaknesses in his knowledge, he studied elliptic. Yet he was also the first at Göttingen to lecture concerning Galois theory, about this time, he became one of the first people to understand the importance of the notion of groups for algebra and arithmetic. In 1858, he began teaching at the Polytechnic school in Zürich, when the Collegium Carolinum was upgraded to a Technische Hochschule in 1862, Dedekind returned to his native Braunschweig, where he spent the rest of his life, teaching at the Institute. He retired in 1894, but did occasional teaching and continued to publish and he never married, instead living with his sister Julia. Dedekind was elected to the Academies of Berlin and Rome, and he received honorary doctorates from the universities of Oslo, Zurich, and Braunschweig. While teaching calculus for the first time at the Polytechnic school, Dedekind developed the now known as a Dedekind cut. The idea of a cut is that an irrational number divides the rational numbers into two classes, with all the numbers of one class being strictly greater than all the numbers of the other class. Every location on the number line continuum contains either a rational or an irrational number, thus there are no empty locations, gaps, or discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind cuts in his pamphlet Stetigkeit und irrationale Zahlen, in modern terminology, Vollständigkeit, Dedekinds theorem states that if there existed a one-to-one correspondence between two sets, then Dedekind said that the two sets were similar. Thus the set N of natural numbers can be shown to be similar to the subset of N whose members are the squares of every member of N, N12345678910
23.
Karl Weierstrass
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Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the father of modern analysis. Despite leaving university without a degree, he studied mathematics and trained as a teacher, eventually teaching mathematics, physics, botany, Weierstrass was born in Ostenfelde, part of Ennigerloh, Province of Westphalia. Weierstrass was the son of Wilhelm Weierstrass, a government official and his interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position, because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his course of study. The outcome was to leave the university without a degree, after that he studied mathematics at the Münster Academy and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city, during this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg, besides mathematics he also taught physics, botanics and gymnastics. Weierstrass may have had a child named Franz with the widow of his friend Carl Wilhelm Borchardt. After 1850 Weierstrass suffered from a period of illness, but was able to publish papers that brought him fame. The University of Königsberg conferred an honorary degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut, which became the Technical University of Berlin. In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which became the Humboldt Universität zu Berlin. He was immobile for the last three years of his life, and died in Berlin from pneumonia, delta-epsilon proofs are first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval, notably, in his 1821 Cours danalyse, Cauchy argued that the limit of continuous functions was itself continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the limit of continuous functions is continuous. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus, using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of functions on closed and bounded intervals
24.
Otto Stolz
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Otto Stolz was an Austrian mathematician noted for his work on mathematical analysis and infinitesimals. Born in Hall in Tirol, he studied in Innsbruck from 1860 and in Vienna from 1863 and his work began with geometry but after the influence of Weierstrass it shifted to real analysis, and many small useful theorems are credited to him. For example, he proved that a function f on a closed interval with midpoint convexity, i. e. f ≤ f + f 2, has left. He died in 1905 shortly after finishing work on Einleitung in die Funktionentheorie and his name lives on in the Stolz-Cesàro theorem. Stolz published a number of papers containing constructions of non-Archimedean extensions of the real numbers and his work, as well as that of Paul du Bois-Reymond, was sharply criticized by Georg Cantor as an abomination. Cantor published a proof-sketch of the inconsistency of infinitesimals, the errors in Cantors proof are analyzed by Ehrlich. Philip Ehrlich The rise of non-Archimedean mathematics and the roots of a misconception, the emergence of non-Archimedean systems of magnitudes, Archive for History of Exact Sciences 60, no. Almanach for 1906, containing obituary OConnor, John J. Robertson, Edmund F. Otto Stolz, MacTutor History of Mathematics archive, Österreich Lexikon, containing Stolzs photograph Haus Der Mathematik
25.
Cauchy sequence
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In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a number of elements of the sequence are less than that given distance from each other. It is not sufficient for each term to become close to the preceding term. For instance, in the harmonic series ∑1 n a difference between consecutive terms decreases as 1 n, however the series does not converge, rather, it is required that all terms get arbitrarily close to each other, starting from some point. More formally, for any given ε >0 there exists an N such that for any m, n > N. The notions above are not as unfamiliar as they might at first appear, the customary acceptance of the fact that any real number x has a decimal expansion is an implicit acknowledgment that a particular Cauchy sequence of rational numbers has the real limit x. In some cases it may be difficult to describe x independently of such a process involving rational numbers. Generalizations of Cauchy sequences in more abstract uniform spaces exist in the form of Cauchy filters, in a similar way one can define Cauchy sequences of rational or complex numbers. Cauchy formulated such a condition by requiring x m − x n to be infinitesimal for every pair of infinite m, n, to define Cauchy sequences in any metric space X, the absolute value |xm - xn| is replaced by the distance d between xm and xn. A metric space X in which every Cauchy sequence converges to an element of X is called complete, the real numbers are complete under the metric induced by the usual absolute value, and one of the standard constructions of the real numbers involves Cauchy sequences of rational numbers. A rather different type of example is afforded by a metric space X which has the discrete metric, any Cauchy sequence of elements of X must be constant beyond some fixed point, and converges to the eventually repeating term. The rational numbers Q are not complete, There are sequences of rationals that converge to irrational numbers, if one considers this as a sequence of real numbers, however, it converges to the real number φ = /2, the Golden ratio, which is irrational. Every Cauchy sequence of numbers is bounded. Every Cauchy sequence of numbers is bounded, hence by Bolzano-Weierstrass has a convergent subsequence, hence is itself convergent. It should be noted, though, that proof of the completeness of the real numbers implicitly makes use of the least upper bound axiom. The alternative approach, mentioned above, of constructing the real numbers as the completion of the rational numbers, makes the completeness of the real numbers tautological. Such a series ∑ n =1 ∞ x n is considered to be convergent if and only if the sequence of sums is convergent. It is a matter to determine whether the sequence of partial sums is Cauchy or not
26.
Metric space
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In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set, a metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces. The most familiar metric space is 3-dimensional Euclidean space, in fact, a metric is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the line segment connecting them. Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, since for any x, y ∈ M, The function d is also called distance function or simply distance. Often, d is omitted and one just writes M for a space if it is clear from the context what metric is used. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations, to be a metric there shouldnt be any one-way roads. The triangle inequality expresses the fact that detours arent shortcuts, many of the examples below can be seen as concrete versions of this general idea. The real numbers with the function d = | y − x | given by the absolute difference. The rational numbers with the distance function also form a metric space. The positive real numbers with distance function d = | log | is a metric space. Any normed vector space is a space by defining d = ∥ y − x ∥. Examples, The Manhattan norm gives rise to the Manhattan distance, the maximum norm gives rise to the Chebyshev distance or chessboard distance, the minimal number of moves a chess king would take to travel from x to y. The British Rail metric on a vector space is given by d = ∥ x ∥ + ∥ y ∥ for distinct points x and y. The name alludes to the tendency of railway journeys to proceed via London irrespective of their final destination, If is a metric space and X is a subset of M, then becomes a metric space by restricting the domain of d to X × X. The discrete metric, where d =0 if x = y and d =1 otherwise, is a simple but important example and this, in particular, shows that for any set, there is always a metric space associated to it. Using this metric, any point is a ball, and therefore every subset is open. A finite metric space is a metric space having a number of points
27.
P-adic number
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The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. In particular, p-adic numbers have the property that they are said to be close when their difference is divisible by a high power of p, the higher the power. P-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this, for example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is derived from the p-adic order. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp and this is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility. The p in p-adic is a variable and may be replaced with a prime or another placeholder variable, the adic of p-adic comes from the ending found in words such as dyadic or triadic. This section is an introduction to p-adic numbers, using examples from the ring of 10-adic numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals, the decadic numbers are generally not used in mathematics, since 10 is not prime, the decadics are not a field. More formal constructions and properties are given below, in the standard decimal representation, almost all real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows 13 =0.333333 …, informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are close to one another. 10-adic numbers use a similar non-terminating expansion, but with a different concept of closeness, whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 4739 and 5739, which differ by 103, are close in the 10-adic world, more precisely, a positive rational number r can be expressed as r =, p/q·10e, where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠0 there exists the maximal e such that this representation is possible, let the 10-adic «absolute value» of r be | r |10, =110 e. Certainly, we have to define |0|10, =0, now, taking p/q =1 and e =0,1,2. We have |100|10 =100, |101|10 = 10−1, |102|10 = 10−2, with the consequence that we have lim + ∞ ← e 10 e =0
28.
Equivalence relation
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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, symmetric 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 then 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, antisymmetric, and transitive. Equality is both a relation and a partial order. Equality is also 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, transitive, 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 also the underlying set for an algebraic structure, and which respects the additional structure