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Unformatted text preview: Solutions of HW#2 : Language of Mathematics Q1. Prove that for any sets A and B , A = ( A B ) ( A B ). Answer Note that you can use Venn diagram to prove the relation. Another way to prove that two sets are equivalent is by proving that each set is a subset of the other as follows. i) To prove that ( A B ) ( A B ) A : Let x ( A B ) ( A B ) x ( A B ) or x ( A B ) ( x A and x / B ) or ( x A and x B ) x A and ( x / B or x B ) x A Thus, all elements of ( A B ) ( A B ) are also elements of A . That is, ( A B ) ( A B ) A (1) ii) To prove that A ( A B ) ( A B ): we follow exactly the same steps but in the reverse order to prove that A ( A B ) ( A B ) (2) From (1) , (2) : A = ( A B ) ( A B ). 1 Q2. Let x and y be integers . Determine whether the following relations are reflexive, sym- metric, antisymmetric, or transitive: i) x y mod 7; ii) xy 1; iii)...
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This homework help was uploaded on 04/19/2008 for the course CS 182 taught by Professor W.szpankowski during the Fall '08 term at Purdue University-West Lafayette.
- Fall '08