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540hw1ans - HW11 1 Prove the second logic distributive law...

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HW1 1 1. Prove the second logic distributive law: [ ° _ ( ± ^ ² )] ° [( ° _ ± ) ^ ( ° _ ² )] : Solution. ° ± ² ± ^ ² ° _ ± ° _ ² [( ° _ ± ) ^ ( ° _ ² )] [ ° _ ( ± ^ ² )] 1 1 1 1 1 1 1 1 1 0 1 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2. Prove De Morgan laws for sets. Hint. You can use (without proof) logic De Morgan laws. Solution. I. [ [ j 2 J A j ] 0 = \ j 2 J A 0 j : ° x 2 [ [ j 2 J A j ] 0 ± ° q ²³ x 2 [ j 2 J A j ´µ ° q ² _ j 2 J ( x 2 A j ) µ First De Morgan logic law ° ² ^ j 2 J q ( x 2 A j ) µ ° ² x 2 \ j 2 J A 0 j µ ; as needed. II. [ \ j 2 J A j ] 0 = [ j 2 J A 0 j : 1 Note to the grader: Please grade the following problems. #1 , #3, #4a, #5b, #6a (20 pts for each problem) 1
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° x 2 [ \ j 2 J A j ] 0 ± q f x 2 [ \ j 2 J A j ] g ° q ² ^ j 2 J ( x 2 A j ) µ Second De Morgan logic law ° ² _ j 2 J q ( x 2 A j ) µ ° ² x 2 [ j 2 J A 0 j µ ; as needed. 3. Let X be a non-empty set and A; B ± X; A; B 6 = ? . Prove (a) A = ( A \ B ) [ ( A \ B 0 ) : (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 0 )] Logic distributive law ° [( x 2 A ) ^ ( x 2 B )] _ [( x 2 A ) ^ ( x 2 B 0 )] f [ x 2 ( A \ B )] _ [ x 2 ( A \ B 0 )] g ° f x 2 [( A \ B ) [ ( A \ B 0 )] 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
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(recall that [( x = 2 B ) ^ ( x 2 B )] ° 0 ), as claimed. Set b X = A [ B . De°ne B 0 = b X n B . Then B 0 = ( A [ B ) n B = A n B . By part (a), A = ( A \ B ) [ ( A \ B 0 ) = ( 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 ) ² ( B 1 [ B 2 ) = ( A 1 ² B 1 ) [ ( A 1 ² B 2 ) [ ( A 2 ² B 1 ) [ ( A 2 ² B 2 ) : (b) ( A 1 \ A 2 ) ² ( B 1 \ B 2 ) = ( A 1 ² B 1 ) \ ( A 2 ² B 2 ) : (c) ( A 1 n A 2 ) ² B = ( A 1 ² B ) n ( A 2 ² B ) : Solution. (a) [( x; y ) 2 ( A 1 [ A 2 ) ² ( B 1 [ B 2 )] ° [( x 2 ( A 1 [ A 2 )) ^ ( y 2 ( B 1 [ B 2 ))] ° f [( x 2 A 1 ) _ ( x 2 A 2 )] g ^ f [( y 2 B 1 ) _ ( y 2 B 2 )] g Logic distributive law ° ff [( x 2 A 1 ) _ ( x 2 A 2 )] g ^ ( y 2 B 1 ) g _ ff [( x 2 A 1 ) _ ( x 2 A 2 )] g ^ ( y 2 B 2 ) g Logic distributive law+commutative laws ° [( x 2 A 1 ) ^ ( y 2 B 1 )] _ [( x 2 A
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