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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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 Fall '08
 W.Szpankowski
 Equivalence relation, Binary relation, Transitive relation, Isomorphism, Preorder, partial order relation

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