This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonempty 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 nonempty 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 ...
View
Full
Document
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.
 Spring '08
 Ruan
 Logic

Click to edit the document details