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Bijection
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== Properties == * A function ''f'': '''R''' โ '''R''' is bijective if and only if its [[graph of a function|graph]] meets every horizontal and vertical line exactly once. * If ''X'' is a set, then the bijective functions from ''X'' to itself, together with the operation of functional composition (<math>\circ</math>), form a [[group (algebra)|group]], the [[symmetric group]] of ''X'', which is denoted variously by S(''X''), ''S<sub>X</sub>'', or ''X''! (''X'' [[factorial]]). * Bijections preserve [[cardinalities]] of sets: for a subset ''A'' of the domain with cardinality |''A''| and subset ''B'' of the codomain with cardinality |''B''|, one has the following equalities: *:|''f''(''A'')| = |''A''| and |''f''<sup>โ1</sup>(''B'')| = |''B''|. *If ''X'' and ''Y'' are [[finite set]]s with the same cardinality, and ''f'': ''X โ Y'', then the following are equivalent: *# ''f'' is a bijection. *# ''f'' is a [[surjection]]. *# ''f'' is an [[injection (mathematics)|injection]]. *For a finite set ''S'', there is a bijection between the set of possible [[total ordering]]s of the elements and the set of bijections from ''S'' to ''S''. That is to say, the number of [[permutation]]s of elements of ''S'' is the same as the number of total orderings of that setโnamely, ''n''!.
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