A bijection, or one-to-one correspondence , is a function which is both injective (or one-to-one) and surjective (or onto). A function has a two-sided inverse exactly when it is a bijection between its domain and range.
Bijections are useful in a variety of contexts. In particular, bijections are frequently used in combinatorics in order to count the elements of a set whose size is unknown. Bijections are also very important in set theory when dealing with arguments concerning infinite sets or in permutation and probability (e.g. 2008 AMC 12B #22).
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