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Enumeration
An ENUMERATION is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and computer science to refer to a listing of all of the elements of a set . The precise requirements for an enumeration (for example, whether the set must be finite , or whether the list is allowed to contain repetitions) depend on the discipline of study and the context of a given problem. Some sets can be enumerated by means of a NATURAL ORDERING (such as 1, 2, 3, 4, ... for the set of positive integers ), but in other cases it may be necessary to impose a (perhaps arbitrary) ordering. In some contexts, such as enumerative combinatorics , the term enumeration is used more in the sense of counting – with emphasis on determination of the number of elements that a set contains, rather than the production of an explicit listing of those elements
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Identity Function
In mathematics , an IDENTITY FUNCTION, also called an IDENTITY RELATION or IDENTITY MAP or IDENTITY TRANSFORMATION, is a function that always returns the same value that was used as its argument. In equations , the function is given by f(x) = x. CONTENTS * 1 Definition * 2 Algebraic property * 3 Properties * 4 See also * 5 References DEFINITIONFormally, if M is a set , the identity function f on M is defined to be that function with domain and codomain M which satisfies f(x) = x for all elements x in M. In other words, the function value f(x) in M (that is, the codomain) is always the same input element x of M (now considered as the domain). The identity function on M is clearly an injective function as well as a surjective function , so it is also bijective . The identity function f on M is often denoted by idM
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Integers
An INTEGER (from the Latin
Latin
integer meaning "whole") is a number that can be written without a fractional component . For example, 21, 4, 0, and −2048 are integers, while 9.75,  5 1⁄2, and √2 are not. The set of integers consists of zero (0 ), the positive natural numbers (1 , 2 , 3 , …), also called whole numbers or counting numbers, and their additive inverses (the NEGATIVE INTEGERS, i.e., −1 , −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold Z {displaystyle mathbb {Z} } (Unicode U+2124 ℤ) standing for the German word Zahlen ( , "numbers"). ℤ is a subset of the sets of rational numbers ℚ, in turn a subset of the real numbers ℝ. Like the natural numbers, ℤ is countably infinite . The integers form the smallest group and the smallest ring containing the natural numbers
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Set Difference
In set theory , the COMPLEMENT of a set A refers to elements not in A. When all sets under consideration are considered to be subsets of a given set U, the ABSOLUTE COMPLEMENT of A is the set of elements in U but not in A. The RELATIVE COMPLEMENT of A with respect to a set B, also termed the DIFFERENCE of sets A and B, written B ∖ A, is the set of elements in B but not in A. CONTENTS* 1 Absolute complement * 1.1 Definition * 1.2 Examples * 1.3 Properties * 2 Relative complement * 2.1 Definition * 2.2 Examples * 2.3 Properties * 3 LaTeX
LaTeX
notation * 4 Complements in various programming languages * 5 See also * 6 Notes * 7 References * 8 External links ABSOLUTE COMPLEMENT The ABSOLUTE COMPLEMENT of A in U: A = U A {displaystyle A^{complement }=Usetminus A} DEFINITIONIf A is a set, then the ABSOLUTE COMPLEMENT of A (or simply the COMPLEMENT OF A) is the set of elements not in A
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Partial Function
In mathematics , a PARTIAL FUNCTION from X to Y (written as f: X ↛ Y) is a function f: X ′ → Y, for some subset X ′ of X. It generalizes the concept of a function f: X → Y by not forcing f to map every element of X to an element of Y (only some subset X ′ of X). If X ′ = X, then f is called a TOTAL FUNCTION and is equivalent to a function. Partial functions are often used when the exact domain , X, is not known (e.g. many functions in computability theory ). Specifically, we will say that for any x ∈ X, either: * f(x) = y ∈ Y (it is defined as a single element in Y) or * f(x) is undefined.For example, we can consider the square root function restricted to the integers g Z Z {displaystyle gcolon mathbb {Z} to mathbb {Z} } g ( n ) = n . {displaystyle g(n)={sqrt {n}}.} Thus g(n) is only defined for n that are perfect squares (i.e., 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined
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Injective
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 function's 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 (a.k.a. bijective function ), which uniquely maps all elements in both domain and codomain to each other (see figures). An injective non-surjective function (injection, not a bijection ) An injective surjective function (bijection ) A non-injective surjective function (surjection , not a bijection ) Occasionally, an injective function from X to Y is denoted f : X ↣ Y, using an arrow with a barbed tail (U+ 21A3 ↣ RIGHTWARDS ARROW WITH TAIL)
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Bijective
In mathematics , a BIJECTION, BIJECTIVE FUNCTION or ONE-TO-ONE CORRESPONDENCE is a function between the elements of two sets , where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets , then the existence of a bijection means they have the same number of elements. For infinite sets the picture is more complicated, leading to the concept of cardinal number , a way to distinguish the various sizes of infinite sets. A bijective function from a set to itself is also called a permutation
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Cardinality
In mathematics , the CARDINALITY of a set is a measure of the "number of elements of the set". For example, the set A = {2, 4, 6} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections , and another which uses cardinal numbers . The cardinality of a set is also called its SIZE, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted  A , with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context . Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A
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Cantor's First Uncountability Proof
GEORG CANTOR\'S FIRST SET THEORY ARTICLE was published in 1874 and contains the first theorems of transfinite set theory , which studies infinite sets and their properties. One of these theorems is "Cantor\'s revolutionary discovery" that the set of all real numbers is uncountably , rather than countably , infinite. This theorem is proved using CANTOR\'S FIRST UNCOUNTABILITY PROOF, which differs from the more familiar proof using his diagonal argument . The title of the article, "ON A PROPERTY OF THE COLLECTION OF ALL REAL ALGEBRAIC NUMBERS" ("Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"), refers to its first theorem: the set of real algebraic numbers is countable. Cantor's article also contains a proof of the existence of transcendental numbers . As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive . Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved
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Well-order
In mathematics , a WELL-ORDER (or WELL-ORDERING or WELL-ORDER RELATION ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a WELL-ORDERED SET. In some academic articles and textbooks these terms are instead written as WELLORDER, WELLORDERED, and WELLORDERING or WELL ORDER, WELL ORDERED, and WELL ORDERING. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element , has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers
Natural numbers
below for an example). In a well-ordered set S, every subset T which has an upper bound has a least upper bound , namely the least element of the subset of all upper bounds of T in S
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Well-ordering
In mathematics , a WELL-ORDER (or WELL-ORDERING or WELL-ORDER RELATION ) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. The set S together with the well-order relation is then called a WELL-ORDERED SET. In some academic articles and textbooks these terms are instead written as WELLORDER, WELLORDERED, and WELLORDERING or WELL ORDER, WELL ORDERED, and WELL ORDERING. Every non-empty well-ordered set has a least element. Every element s of a well-ordered set, except a possible greatest element , has a unique successor (next element), namely the least element of the subset of all elements greater than s. There may be elements besides the least element which have no predecessor (see Natural numbers
Natural numbers
below for an example). In a well-ordered set S, every subset T which has an upper bound has a least upper bound , namely the least element of the subset of all upper bounds of T in S
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Injection (mathematics)
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 function's 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 (a.k.a. bijective function ), which uniquely maps all elements in both domain and codomain to each other (see figures). An injective non-surjective function (injection, not a bijection ) An injective surjective function (bijection ) A non-injective surjective function (surjection , not a bijection ) Occasionally, an injective function from X to Y is denoted f : X ↣ Y, using an arrow with a barbed tail (U+ 21A3 ↣ RIGHTWARDS ARROW WITH TAIL)
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Decidable Language
In mathematics , logic and computer science , a formal language (a set of finite sequences of symbols taken from a fixed alphabet ) is called RECURSIVE if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a total Turing machine
Turing machine
(a Turing machine
Turing machine
that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called DECIDABLE. The concept of DECIDABILITY may be extended to other models of computation . For example one may speak of languages decidable on a non-deterministic Turing machine
Turing machine
. Therefore, whenever an ambiguity is possible, the synonym for "recursive language" used is TURING-DECIDABLE LANGUAGE, rather than simply decidable
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First Uncountable Ordinal
In mathematics, the FIRST UNCOUNTABLE ORDINAL, traditionally denoted by ω1 or sometimes by Ω, is the smallest ordinal number that, considered as a set , is uncountable . It is the supremum (least upper bound) of all countable ordinals. The elements of ω1 are the countable ordinals, of which there are uncountably many. Like any ordinal number (in von Neumann's approach), ω1 is a well-ordered set , with set membership ("∈") serving as the order relation. ω1 is a limit ordinal , i.e. there is no ordinal α with α + 1 = ω1. The cardinality of the set ω1 is the first uncountable cardinal number , ℵ1 (aleph-one ). The ordinal ω1 is thus the initial ordinal of ℵ1. Indeed, in most constructions ω1 and ℵ1 are equal as sets. To generalize: if α is an arbitrary ordinal we define ωα as the initial ordinal of the cardinal ℵα. The existence of ω1 can be proven without the axiom of choice
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