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  2. Unordered pair - Wikipedia

    en.wikipedia.org/wiki/Unordered_pair

    In mathematics, an unordered pair or pair set is a set of the form {a, b}, i.e. a set having two elements a and b with no particular relation between them, where {a, b} = {b, a}. In contrast, an ordered pair ( a , b ) has a as its first element and b as its second element, which means ( a , b ) ≠ ( b , a ).

  3. Kripke–Platek set theory - Wikipedia

    en.wikipedia.org/wiki/Kripke–Platek_set_theory

    Theorem: If A and B are sets, then there is a set A×B which consists of all ordered pairs (a, b) of elements a of A and b of B. Proof: The singleton set with member a, written {a}, is the same as the unordered pair {a, a}, by the axiom of extensionality. The singleton, the set {a, b}, and then also the ordered pair

  4. List of order structures in mathematics - Wikipedia

    en.wikipedia.org/wiki/List_of_order_structures...

    Lattices, partial orders in which each pair of elements has a greatest lower bound and a least upper bound. Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be

  5. Axiom of pairing - Wikipedia

    en.wikipedia.org/wiki/Axiom_of_pairing

    The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, and thus it is sometimes omitted.

  6. Implementation of mathematics in set theory - Wikipedia

    en.wikipedia.org/wiki/Implementation_of...

    In NFU, all the set definitions given work by stratified comprehension; in ZFC, the existence of the unordered pair is given by the Axiom of Pairing, the existence of the empty set follows by Separation from the existence of any set, and the binary union of two sets exists by the axioms of Pairing and Union (= {,}).

  7. Pair - Wikipedia

    en.wikipedia.org/wiki/Pair

    Mathematics. 2 (number), two of something, a pair; Unordered pair, or pair set, in mathematics and set theory; Ordered pair, or 2-tuple, in mathematics and set theory; Pairing, in mathematics, an R-bilinear map of modules, where R is the underlying ring; Pair type, in programming languages and type theory, a product type with two component types

  8. Set theory - Wikipedia

    en.wikipedia.org/wiki/Set_theory

    Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects.Although objects of any kind can be collected into a set, set theory — as a branch of mathematics — is mostly concerned with those that are relevant to mathematics as a whole.

  9. Room square - Wikipedia

    en.wikipedia.org/wiki/Room_square

    Each cell of the array is either empty or contains an unordered pair from the set of symbols; Each symbol occurs exactly once in each row and column of the array; Every unordered pair of symbols occurs in exactly one cell of the array. An example, a Room square of order seven, if the set of symbols is integers from 0 to 7: