Difference between revisions of "Bijection"
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If there is a bijection between two sets, then we say that they have the same [[cardinality]], or size. | If there is a bijection between two sets, then we say that they have the same [[cardinality]], or size. | ||
| − | Bijections between [[infinite]] sets are particularly interesting, as they produce some counterintuitive results. For instance, there is a bijection between the integers and the even integers that maps every integer x to the even integer 2x. Notice that this satisfies the definition of bijection, | + | Bijections between [[infinite]] sets are particularly interesting, as they produce some counterintuitive results. For instance, there is a bijection between the integers and the even integers that maps every integer x to the even integer 2x. Notice that this satisfies the definition of bijection, even though it would then seem to imply that the integers and the even integers have the same size. However, since both sets are of infinite size, it merely illustrates the idea that "infinity times two is infinity". |
The existence of a bijection from one set to the other also implies that a converse bijection exists. | The existence of a bijection from one set to the other also implies that a converse bijection exists. | ||
Revision as of 13:01, December 12, 2008
A bijection is a one-to-one, onto mapping between two sets. In other words, a bijection between sets A and B is a mapping such that every element in the A is mapped to a distinct element in B, and every element in the B has a distinct element in the A mapped to it. For example, one bijection between the sets {A, B, C} and {1, 2, 3}, maps: A to 1, B to 2, and 3 to C.
If there is a bijection between two sets, then we say that they have the same cardinality, or size.
Bijections between infinite sets are particularly interesting, as they produce some counterintuitive results. For instance, there is a bijection between the integers and the even integers that maps every integer x to the even integer 2x. Notice that this satisfies the definition of bijection, even though it would then seem to imply that the integers and the even integers have the same size. However, since both sets are of infinite size, it merely illustrates the idea that "infinity times two is infinity".
The existence of a bijection from one set to the other also implies that a converse bijection exists.