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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
In mathematics, and more specifically in order theory, several different types of ordered set have been studied. They include: Cyclic orders, orderings in which triples of elements are either clockwise or counterclockwise; Lattices, partial orders in which
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
The graph shown has one maximum clique, the triangle {1,2,5}, and four more maximal cliques, the pairs {2,3}, {3,4}, {4,5}, and {4,6}. An undirected graph is formed by a finite set of vertices and a set of unordered pairs of vertices, which are called edges.
The first textbook on graph theory was written by Dénes Kőnig, and published in 1936. [26] Another book by Frank Harary , published in 1969, was "considered the world over to be the definitive textbook on the subject", [ 27 ] and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.