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Unformatted text preview: Math 100 Cardinality Martin H. Weissman 1. Cardinality of Sets Suppose that A and B are finite sets . Recall that we have proven the following theorems, relating the number of elements of A , the number of elements of B , and the existence of functions from A to B : • # A ≤ # B if and only if there exists an injective function from A to B . • # A ≥ # B if and only if there exists a surjective function from A to B . • # A = # B if and only if there exists a bijective function from A to B . The study of “infinity” in set theory begins with the following important observation: even though the “number” # A does not yet have a definition when A is an infinite set, the existence of injective, surjective, and bijective functions makes sense for infinite sets. From this observation, we turn the above theorems into definitions , when A and B are arbitrary (not necessarily finite) sets: Definition 1. Suppose that A and B are sets. We say that # A ≤ # B if there exists an injective function from A to B . We say that # A ≥ # B if there exists a surjective function from A to B . We say that # A = # B if there exists a bijective function from A to B . In this new formulation, # A is not a number, in the traditional sense. Rather, # A is a cardinal number – it is used to compare A to other sets with the relations =, ≤ , and ≥ . For example, # Z is not a number in the traditional sense (it is an infinite cardinal number), but we can still compare cardinalities . For example, # Z ≤ # R , since there is an injective map from Z to R . Similarly, we may see that # Q ≥ # N , by constructing a surjective map from Q to N . To construct such a map, note that every rational number can be expressed as a quotient of integers p/q – by reducing p/q into...
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.
- Fall '08