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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
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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
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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