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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
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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
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Cantor's First Uncountability Proof
Georg Cantor's first set theory article
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.[1] One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.[2] 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
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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
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Partial Function
In mathematics, a partial function from X to Y (written as f: X ↛ Y or 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[example needed]). 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
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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.Contents1 Definition 2 Algebraic property 3 Properties 4 See also 5 ReferencesDefinition[edit] Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfiesf(x) = x   for all elements x in M.[1]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)
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Integers
An integer (from the Latin
Latin
integer meaning "whole")[note 1] 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,[1][2] 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
Unicode
U+2124 ℤ) standing for the German word Zahlen ([ˈtsaːlən], "numbers").[3][4] Z is a subset of the set of all 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.Contents1 Absolute complement1.1 Definition 1.2 Examples 1.3 Properties2 Relative complement2.1 Definition 2.2 Examples 2.3 Properties3 LaTeX
LaTeX
notation 4 Complements in various programming languages 5 See also 6 Notes 7 References 8 External linksAbsolute complement[edit]The absolute complement of A in U: A ∁ = U ∖ A displaystyle A^ complement =Usetminus A Definition[edit] If 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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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] 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, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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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)
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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
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Initial Segment
In mathematics, an upper set (also called an upward closed set or just an upset) of a partially ordered set (X,≤) is a subset U with the property that, if x is in U and x≤y, then y is in U. The dual notion is lower set (alternatively, down set, decreasing set, initial segment, semi-ideal; the set is downward closed), which is a subset L with the property that, if x is in L and y≤x, then y is in L. The terms order ideal or ideal are sometimes used as synonyms for lower set.[1][2][3] This choice of terminology fails to reflect the notion of an ideal of a lattice because a lower set of a lattice is not necessarily a sublattice.[1]Contents1 Properties 2 Ordinal numbers 3 See also 4 ReferencesProperties[edit]Every partially ordered set is an upper set of itself. The intersection and the union of upper sets is again an upper set. The complement of any upper set is a lower set, and vice versa. Given a partially ordered set (X,≤), the family of lower sets of
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Ordinal Number
In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order. An ordinal number is used to describe the order type of a well ordered set (though this does not work for a well ordered proper class)
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Transfinite Induction
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Let P(α) be a property defined for all ordinals α. Suppose that whenever P(β) is true for all β < α, then P(α) is also true. Then transfinite induction tells us that P is true for all ordinals. Usually the proof is broken down into three cases:Zero case: Prove that P(0) is true. Successor case: Prove that for any successor ordinal α+1, P(α+1) follows from P(α) (and, if necessary, P(β) for all β < α). Limit case: Prove that for any limit ordinal λ, P(λ) follows from [P(β) for all β < λ].All three cases are identical except for the type of ordinal considered. They do not formally need to be considered separately, but in practice the proofs are typically so different as to require separate presentations
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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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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)
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