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sols10 - 10. OCTOBER 5TH: FUNCTIONS. 25 10. October 5th:...

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Unformatted text preview: 10. OCTOBER 5TH: FUNCTIONS. 25 10. October 5th: Functions. 10.1 . Let A , B be sets, and f : A B a function. Let S and T be subsets of A . Is f ( S T ) necessarily equal to f ( S ) f ( T )? Give a proof or find a coun- terexample. Is f ( S T ) necessarily equal to f ( S ) f ( T )? Proof or counterexample. Answer: f ( S T ) and f ( S ) f ( T ) need not be equal. For example, consider the function f : { , *} { } defined by f ( ) = 0, f ( * ) = 0, and let S = {} , T = {*} . Then S T = , so f ( S T ) = ; while f ( S ) f ( T ) = { } = . Note that the inclusion f ( S T ) f ( S ) f ( T ) always holds. (Why?) It is true that f ( S ) f ( T ) = f ( S T ) for all A, B, f, S, T . Proof: Since S and T are subsets of S T , then f ( S ) and f ( T ) are both subsets of f ( S T ); this shows f ( S ) f ( T ) f ( S T ). On the other hand, if b f ( S T ) then a S T such that f ( a ) = b . Since a S T , then a S or a T . If a S , then...
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sols10 - 10. OCTOBER 5TH: FUNCTIONS. 25 10. October 5th:...

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