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hw5 - CSE 20 Discrete Mathematics Fall 2011 Problem Set 5...

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CSE 20: Discrete Mathematics Fall 2011 Problem Set 5 Instructor: Daniele Micciancio Due on: Fri. Nov. 11, 2011 Problem 1 (8 points) Remember, the image of a set X A under a function f : A B is the set f ( X ) = { f ( x ) | x X } , while the inverse image of a set Y B is the set f - 1 ( Y ) = { x A | f ( x ) Y } . In class we asked if for every set X A and function f : A B , it is true that f - 1 ( f ( X )) = X , and we began answering this question by proving that X f - 1 ( f ( X )). We left open question whether the reverse inclusion X f - 1 ( f ( X )) also holds. Prove or disprove that for every f : A B and X A , it holds that X f - 1 ( f ( X )) also holds. Your answer should contain: 1. A clear claim of which statement you are proving. (This can be the given statement or its negation. 2. A proof that your claim is correct. Problem 2 (16 points) In this problem you are asked to prove that for any function f : A B and set Y B , if f is onto, then f ( f - 1 ( Y )) = Y . Structure your proof as follows: Prove as a first lemma that f ( f - 1 ( Y )) Y .
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