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Given a set A, the identity function on A is a bijection from A to itself, showing that every set A is equinumerous to itself: A ~ A. Symmetry For every bijection between two sets A and B there exists an inverse function which is a bijection between B and A, implying that if a set A is equinumerous to a set B then B is also equinumerous to A: A ...
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
The set of all equivalence classes in with respect to an equivalence relation is denoted as /, and is called modulo (or the quotient set of by ). [3] The surjective map x ↦ [ x ] {\displaystyle x\mapsto [x]} from X {\displaystyle X} onto X / R , {\displaystyle X/R,} which maps each element to its equivalence class, is called the canonical ...
Given any set , an equivalence relation over the set [] of all functions can be obtained as follows. Two functions are deemed equivalent when their respective sets of fixpoints have the same cardinality, corresponding to cycles of length one in a permutation.
Two metrics and on X are strongly or bilipschitz equivalent or uniformly equivalent if and only if there exist positive constants and such that, for every ,, (,) (,) (,).In contrast to the sufficient condition for topological equivalence listed above, strong equivalence requires that there is a single set of constants that holds for every pair of points in , rather than potentially different ...
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
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