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Assignment2 - Math 240 Fall 2009 Assignment 2 Due Monday...

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Math 240, Fall 2009 — Assignment 2 — Due: Monday, October 5 1. Suppose that five ones and four zeros are arranged around a circle. Let’s use the word bit to mean either a zero or a one. We do the following steps: Step 1: Between any two equal bits, insert a 0 and between any two unequal bits, insert a 1. This produces nine new bits. Step 2: Erase the nine original bits. Prove that when we perform this procedure repeatedly, we never have nine zeros around the circle after Step 2. Hint: Give a proof by contradiction. Work backward, assuming that you did end up with nine zeros. 2. Suppose that A , B , C are sets. Prove that ( B - A ) ( C - A ) = ( B C ) - A . 3. Suppose A and B are two finite sets. Recall that P ( A ) is the power set of A , which is the set of all subsets of A . (a) Prove, or disprove, that P ( A B ) = P ( A ) P ( B ). (b) Prove, or disprove, that P ( A B ) = P ( A ) P ( B ). 4. Each of the following functions f is a function from R to R . Determine whether each f is injective (one-to-one), surjective (onto), or both. Please give reasons. (a) f ( x ) = - 3 x + 4 (b) f ( x ) = - 3 x 2 + 7 (c) f ( x ) =
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