Difference between revisions of "Bijection"
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| − | A '''bijection''', or ''one-to-one correspondence '', is a [[function]] which is both [[injection|injective]] ( | + | A '''bijection''', or '''one-to-one correspondence ''', is a [[function]] which is both [[injection|injective]] (''one-to-one'') and [[surjection|surjective]] (''onto''). A function has a [[Function#Inverses|two-sided inverse]] exactly when it is a bijection between its [[domain]] and [[range]]. |
| − | Bijections are useful in a variety of contexts. | + | 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. |
| − | {{ | + | == Problems == |
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| + | * [[2008 AMC 12B Problems/Problem 22]] | ||
| + | * [[2001 AIME I Problems/Problem 6]] | ||
| + | * [[2006 AIME II Problems/Problem 4]] | ||
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| + | This is recommended to be learned around the time you are introduced to the [[ball-and-urn]] method, so that you can become increasingly familiar with the more advanced concepts of [[combinatorics]]. | ||
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| + | {{stub}} | ||
Latest revision as of 21:12, 20 March 2025
A bijection, or one-to-one correspondence , is a function which is both injective (one-to-one) and surjective (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.
Problems
This is recommended to be learned around the time you are introduced to the ball-and-urn method, so that you can become increasingly familiar with the more advanced concepts of combinatorics.
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