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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]
The existence axioms, e.g. the existence of the unordered pair, is also implemented indirectly by the definition of term constructors. The system includes equality, the membership predicate and the following standard definitions: Singleton: A set with one member; Unordered pair: A set with two distinct members.
The ordered pair (a, b) is different from the ordered pair (b, a), unless a = b. In contrast, the unordered pair, denoted {a, b}, equals the unordered pair {b, a}. Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.
The set of all ordered pairs obtained from two sets, where each pair consists of one element from each set. cardinal 1. A cardinal number is an ordinal with more elements than any smaller ordinal cardinality The number of elements of a set categorical 1. A theory is called categorical if all models are isomorphic. This definition is no longer ...
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
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).
Develop: Unordered and ordered pairs, relations, functions, domain, range, function composition. V. Substitution: If f is a [class] function and domain f is a set, then range f is a set. The import of V is that of the axiom schema of replacement in NBG and ZFC. VI. Amalgamation: If x is a set, then is a set.
{{,},}, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with two distinct vertices). To avoid ambiguity, this type of object may be called precisely an undirected simple graph.