Difference between revisions of "Bijection"
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A '''bijection''', or ''one-to-one correspondence '', is a [[function]] which is both [[injection|injective]] (or ''one-to-one'') and [[surjection|surjective]] (or ''onto''). A function has a [[Function#The_Inverse_of_a_Function|two-sided inverse]] exactly when it is a bijection between its [[domain]] and [[range]]. | A '''bijection''', or ''one-to-one correspondence '', is a [[function]] which is both [[injection|injective]] (or ''one-to-one'') and [[surjection|surjective]] (or ''onto''). A function has a [[Function#The_Inverse_of_a_Function|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 | + | 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. |
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| + | <math>\textbf{\underline{Examples:}}</math> | ||
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| + | [[2008 AMC 12B Problems/Problem 22]] | ||
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| + | [[2001 AIME I Problems/Problem 6]] | ||
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| + | [[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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Latest revision as of 02:39, 29 November 2021
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.
2008 AMC 12B Problems/Problem 22
2001 AIME I Problems/Problem 6
2006 AIME II Problems/Problem 4
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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