Difference between revisions of "Correspondence"
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| − | ( | + | A '''one-to-one correspondence''', or ''bijection'', is a function which is both [[injection|injective]] (or ''one-to-one'') and [[surjection|surjective]] (or ''onto''). A function is [[Function#The_Inverse_of_a_Function|invertible]] exactly when it is a bijection. |
| − | + | One-to-one correspondences 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. | |
== Examples == | == Examples == | ||
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=464897#p464897 AIME 2006I/4] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=464897#p464897 AIME 2006I/4] | ||
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=384179#p384179 AIME 2001I/6] | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=384179#p384179 AIME 2001I/6] | ||
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Revision as of 16:25, 29 June 2006
A one-to-one correspondence, or bijection, is a function which is both injective (or one-to-one) and surjective (or onto). A function is invertible exactly when it is a bijection.
One-to-one correspondences 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.
