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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 ).
The points of the Cremona–Richmond configuration may be identified with the = unordered pairs of elements of a six-element set; these pairs are called duads.Similarly, the lines of the configuration may be identified with the 15 ways of partitioning the same six elements into three pairs; these partitions are called synthemes.
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
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
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.
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
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
The configuration space of all unordered pairs of points on the circle is the Möbius strip. In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space.