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An unordered pair is a finite set; its cardinality (number of elements) is 2 or (if the two elements are not distinct) 1. In axiomatic set theory, the existence of unordered pairs is required by an axiom, the axiom of pairing. More generally, an unordered n-tuple is a set of the form {a 1, a 2,... a n}. [5] [6] [7]
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; Topological pair, an inclusion of topological spaces
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.
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:
2 Unordered pairs and binary sets are different and distinct concepts
In this setting, an edge is an unordered pair of vertices, and a two-edge graph is an unordered pair of edges. The number of possible edges is a triangular number, and the number of pairs of edges (allowing both edges to connect the same two vertices) is a doubly triangular number. [4]
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A name–value pair, also called an attribute–value pair, key–value pair, or field–value pair, is a fundamental data representation in computing systems and applications. Designers often desire an open-ended data structure that allows for future extension without modifying existing code or data.