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( Cardinality)
In mathematics, the cardinality of a set is a measure of the "number of elements of the set". For example, the set A = {1, 2, 3} contains 3 elements, and therefore A has a cardinality of 3. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. Two sets A and B have the same cardinality if there exists a bijection, that is, an injective and surjective function, from A to B. For example, the set E = {2, 4, 6, ...} of positive even numbers has the same cardinality as the set N = {1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. A set A has cardinality greater than or equal to the cardinality of B if there exists an injective function from B into A. The set A has cardinality strictly greater than the cardinality of B,if there is an injective function, but no bijective function, from B to A. For example, the set R of all real numbers has cardinality strictly greater than the cardinality of the set N of all natural numbers, because the inclusion map i&_160; N ? R is injective, but it can be shown that there does not exist a bijective function from N to R. Above, "cardinality" was defined functionally. That is, the "cardinality" of a set was not defined as a specific object itself. However, such an object can be defined as follows.
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Cardinality Subcategories
Cardinality Articles
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