540hw1ans - HW1 1 1 Prove the second logic distributive...

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Unformatted text preview: HW1 1 1. Prove the second logic distributive law: [ & _ ( ¡ ^ ¢ )] & [( & _ ¡ ) ^ ( & _ ¢ )] : Solution. & ¡ ¢ ¡ ^ ¢ & _ ¡ & _ ¢ [( & _ ¡ ) ^ ( & _ ¢ )] [ & _ ( ¡ ^ ¢ )] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2. Prove De Morgan laws for sets. Hint. You can use (without proof) logic De Morgan laws. Solution. I. [ [ j 2 J A j ] = \ j 2 J A j : & x 2 [ [ j 2 J A j ] ¡ & q ¢£ x 2 [ j 2 J A j ¤¥ & q ¢ _ j 2 J ( x 2 A j ) ¥ First De Morgan logic law & ¢ ^ j 2 J q ( x 2 A j ) ¥ & ¢ x 2 \ j 2 J A j ¥ ; as needed. II. [ \ j 2 J A j ] = [ j 2 J A j : 1 Note to the grader: Please grade the following problems. #1 , #3, #4a, #5b, #6a (20 pts for each problem) 1 & x 2 [ \ j 2 J A j ] ¡ q f x 2 [ \ j 2 J A j ] g & q ¢ ^ j 2 J ( x 2 A j ) £ Second De Morgan logic law & ¢ _ j 2 J q ( x 2 A j ) £ & ¢ x 2 [ j 2 J A j £ ; as needed. 3. Let X be a non-empty set and A;B ¡ X; A;B 6 = ? . Prove (a) A = ( A \ B ) [ ( A \ B ) : (b) A = ( A n B ) [ ( A \ B ) . Solution. (a) Indeed, ( x 2 A ) & ( x 2 A ) ^ [( x 2 B ) _ q ( x 2 B )] & ( x 2 A ) ^ [( x 2 B ) _ ( x 2 B )] Logic distributive law & [( x 2 A ) ^ ( x 2 B )] _ [( x 2 A ) ^ ( x 2 B )] f [ x 2 ( A \ B )] _ [ x 2 ( A \ B )] g & f x 2 [( A \ B ) [ ( A \ B )] g ; as needed. (b) First we prove an auxiliary statement: A n B = ( A [ B ) n B: Indeed, [ x 2 ( A [ B ) n B ] & ( x 2 ( A [ B )) ^ q ( x 2 B ) & ( x 2 ( A [ B )) ^ ( x = 2 B ) Commutative logic law & ( x = 2 B ) ^ [( x 2 A ) _ ( x 2 B )] Distributive logic law & [( x = 2 B ) ^ ( x 2 A )] _ [( x = 2 B ) ^ ( x 2 B )] & [( x 2 A ) ^ ( x = 2 B )] & [ x 2 A n B ] 2 (recall that [( x = 2 B ) ^ ( x 2 B )] & ), as claimed. Set b X = A [ B . De&ne B = b X n B . Then B = ( A [ B ) n B = A n B . By part (a), A = ( A \ B ) [ ( A \ B ) = ( A \ B ) [ ( A n B ) , as needed. 4. Let A 1 ;A 2 ;B 1 ;B 2 and B be non-empty sets. Prove: (a) ( A 1 [ A 2 ) ¡ ( B 1 [ B 2 ) = ( A 1 ¡ B 1 ) [ ( A 1 ¡ B 2 ) [ ( A 2 ¡ B 1 ) [ ( A 2 ¡ B 2 ) : (b) ( A 1 \ A 2 ) ¡ ( B 1 \ B 2 ) = ( A 1 ¡ B 1 ) \ ( A 2 ¡ B 2 ) : (c) ( A 1 n A 2 ) ¡ B = ( A 1 ¡ B ) n (...
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This note was uploaded on 02/16/2012 for the course MATH 540 taught by Professor Ruan during the Spring '08 term at University of Illinois, Urbana Champaign.

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540hw1ans - HW1 1 1 Prove the second logic distributive...

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