WA 10 - Nicola Hodges David Walnut MATH 290-002 December...

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David Walnut MATH 290-002 December 14 th , 2010 Writing Assignment 10 1. Let A and B be finite sets with A~B. Suppose f: A B a. If f is one-to-one, show that f is onto B. Let A and B be two finite Sets and A B. Assume f is a function from A to B. Assume f is one to one, we will show that f is onto. Assume f is not not onto, we will show f is not one-to-one. Since A is finite, A N k and A contains k elements. Since B is finite B N k and since f is not onto b, there exists element b such that b is not in the range of A. Since A N k and B N k, then B-{b} N k. Define Define b. If f is onto B, prove that f is one-to-one. Let A and B be two finite Sets. Assume f is a function from A to B. Assume function f is onto B, we will show f is one-to-one. Since f is onto, for each b B, there exists a A such that f(a) = b. Therefore each element b of B has a preimage in A. Since we assumed A B, then the cardinality of A equals the cardinality of B, Hence, each element of B has only one preimage in A. Therefore, different elements of A have different images in
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WA 10 - Nicola Hodges David Walnut MATH 290-002 December...

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