540hw1 - HW1 1. Prove the second logic distributive law: [...

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Unformatted text preview: HW1 1. Prove the second logic distributive law: [ _ ( ^ )] [( _ ) ^ ( _ )] : Hint: Fill up the following truth table. 1 1 0 1 1 0 0 0 1 0 ? ? ? ? ? 0 1 1 ? ? ? ? ? 0 ? ? ? ? ? ? ? ? ^ ? ? ? ? ? ? ? ? _ ? ? ? ? ? ? ? ? _ [( _ ) ^ ( _ )] ? ? ? ? ? ? ? ? [ _ ( ^ )] ? ? ? ? ? ? ? ? 2. Prove generalized De Morgan laws for a (possibly in…nite) family Aj 2 2X j j 2 J of sets. Hint. You can use (without proof) logic (generalized) De Morgan laws. 3. Let X be a non-empty set and A; B logic laws the following (a) A = (A \ B ) [ (A \ B 0 ) : (b) A = (AnB ) [ (A \ B ). X; A; B 6= ?. Prove by using 4. Let A1 ; A2 ; B1 ; B2 ,B 6= ? and A1 nA2 6= ?. Prove: (a) (A1 [ A2 ) (B1 [ B2 ) = (A1 B1 ) [ (A1 B2 ) [ (A2 B1 ) [ (A2 (b) (A1 \ A2 ) (B1 \ B2 ) = (A1 B1 ) \ (A2 B2 ) : 1 B2 ) : (c) Hint: Use (A1 nA2 ) B = (A1 [(x; y ) 2 X B ) n (A2 B) : [(x 2 X ) ^ (y 2 Y )] : Y] 5. Prove: (a) f ([j 2J Aj ) = [j 2J f (Aj ). (b) f (\j 2J Aj ) \j 2J f (Aj ); give an example showing that the equality sign need not hold. (c) f (A) nf (B ) f (AnB ); give an example showing that the equality sign need not hold. (d) f 1 ([j 2J Aj ) = [j 2J f 1 (Aj ). (e) f 1 (\j 2J Aj ) = \j 2J f 1 (Aj ). (f) f 1 (AnB ) = f 1 (A) nf 1 (B ). 6. Let X be a set and A; B; C; E 2 2X . Recall that A 4 B is called the symmetric di¤erence of the sets A and B : A 4 B = (AnB ) [ (B nA) = (A [ B ) n (A \ B ) : For a set A 2 2X , set A0 = X nA. Prove: (a) A 4 B = B 4 A and A 4 (B 4 C ) = (A 4 B ) 4 C . (b) A 4 B = ? if and only if A = B: (c) A 4 B = X if and only if A = B 0 . (d) A 4 ? = A and A 4 X = A0 . (e) (A 4 B ) \ E = (A \ E ) 4 (B \ E ). 7. Let A; B; C 2 2X . Prove: (A 4 B ) 4 (B 4 C ) = A 4 C: 8. For a non-empty set A, idA always denotes the mapping de…ned by idA (a) = a 8a 2 A. Prove that f : X ! Y is an injective mapping if and only if 9 g : Y ! X such that g f = idX (i.e., f is injective, f has a left inverse). 2 ...
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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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540hw1 - HW1 1. Prove the second logic distributive law: [...

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