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30 relations: Axiom of choice, Binomial coefficient, Cardinal number, Cardinality, Cardinality of the continuum, Club set, Cofinal (mathematics), Continuum hypothesis, Directed set, Disjoint union, Greatest and least elements, Idempotence, Image (mathematics), König's theorem (set theory), Limit ordinal, Mathematics, Maximal and minimal elements, Net (mathematics), Order isomorphism, Order theory, Order type, Ordinal number, Partially ordered set, Real number, Regular cardinal, Subsequence, Successor ordinal, Total order, Von Neumann cardinal assignment, Well-order.
Axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty.
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Binomial coefficient
In mathematics, any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient.
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Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets.
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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
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Cardinality of the continuum
In set theory, the cardinality of the continuum is the cardinality or “size” of the set of real numbers \mathbb R, sometimes called the continuum.
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Club set
In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded (see below) relative to the limit ordinal.
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Cofinal (mathematics)
In mathematics, let A be a set and let ≤ be a binary relation on A. Then a subset B of A is said to be cofinal if it satisfies the following condition: This definition is most commonly applied when A is a partially ordered set or directed set under the relation ≤. Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of “subsequence”.
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Continuum hypothesis
In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets.
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Directed set
In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty set A together with a reflexive and transitive binary relation ≤ (that is, a preorder), with the additional property that every pair of elements has an upper bound.
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Disjoint union
In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.
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Greatest and least elements
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S that is greater than every other element of S. The term least element is defined dually, that is, it is an element of S that is smaller than every other element of S. Formally, given a partially ordered set (P, ≤), an element g of a subset S of P is the greatest element of S if Hence, the greatest element of S is an upper bound of S that is contained within this subset.
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Idempotence
Idempotence is the property of certain operations in mathematics and computer science that they can be applied multiple times without changing the result beyond the initial application.
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
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König's theorem (set theory)
In set theory, König's theorem states that if the axiom of choice holds, I is a set, \kappa_i and \lambda_i are cardinal numbers for every i in I, and \kappa_i for every i in I, then The sum here is the cardinality of the disjoint union of the sets mi, and the product is the cardinality of the Cartesian product.
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Limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maximal and minimal elements
In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.
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Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
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Order isomorphism
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets).
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Order theory
Order theory is a branch of mathematics which investigates the intuitive notion of order using binary relations.
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Order type
In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X → Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in the correct order).
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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.
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Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality.
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Subsequence
In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.
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Successor ordinal
In set theory, the successor of an ordinal number α is the smallest ordinal number greater than α.
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Total order
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
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Von Neumann cardinal assignment
The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers.
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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.
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References
[1] https://en.wikipedia.org/wiki/Cofinality
