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Unformatted text preview: 13. OCTOBER 13TH: PROOF OF THEOREM ?? 31 13. October 13th: Proof of Theorem 12.11 13.1 . Suppose A and B are finite sets with the same number of elements, and that f : A B is an injective function. Prove that f is also surjective. Now suppose that A and B are both infinite sets, and that f : A B is an injective function. Is f necessarily surjective? Either give a proof, or a counterexam- ple. Answer: If A is a finite set with n elements and f : A B is an injective function, then the image im f of f also consists of n elements. Indeed, there cant be more than n elements in im f , since each element of im f is the image of some element of A ; and there cant be fewer than n , because otherwise two different elements of A would have the same image, and this would contradict the injectivity of f . Thus, im f is a subset of B with n elements. If B also has n elements, as specified in this problem, then im f must be the whole of B . This says that every element of....
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This note was uploaded on 12/14/2011 for the course MGF 3301 taught by Professor Aluffi during the Fall '11 term at FSU.
- Fall '11