Homework 3 Solutions W11

Homework 3 - EECS 203 Homework 3 Solutions Section 2.1 1(E 8defg d True e True f True g False since{∅{∅ ={∅ 2(E 22 a No since a power set

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Unformatted text preview: EECS 203: Homework 3 Solutions Section 2.1 1. (E) 8defg d) True e) True f) True g) False, since {{∅} , {∅}} = {{∅}} . 2. (E) 22 a) No, since a power set should contains at least the empty set ∅ ; note that P ( ∅ ) = {∅} . b) Yes, P ( { a } ) = {∅ , { a }} . c) No, since the set has only three elements; note that | P ( S ) | = 2 | S | for any set S . d) Yes, P ( { a,b } ) = {∅ , { a } , { b } , { a,b }} . 3. (E) 28cd c) C × A × B = { (0 ,a,x ) , (0 ,a,y ) , (0 ,b,x ) , (0 ,b,y ) , (0 ,c,x ) , (0 ,c,y ) , (1 ,a,x ) , (1 ,a,y ) , (1 ,b,x ) , (1 ,b,y ) , (1 ,c,x ) , (1 ,c,y ) } d) B × B × B = { ( x,x,x ) , ( x,x,y ) , ( x,y,x ) , ( x,y,y ) , ( y,x,x ) , ( y,x,y ) , ( y,y,x ) , ( y,y,y ) } 4. (E) 36bc b) The truth set is the empty set ∅ . c) The truth set contains all integers except and 1 , i.e., { ...,- 3 ,- 2 ,- 1 , 2 , 3 ,... } . Section 2.2 5. (M) 18de (Prove part d by using Membership Tables. Prove part e by showing that each side is a subset of the other side.) d) Use the membership table: A B C A- C C- B ( A- C ) ∩ ( C- B ) ∅ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Column 6 and 7 are exactly the same. Therefore, ( A- C ) ∩ ( C- B ) = ∅ . e) To show that ( B- A ) ∪ ( C- A ) = ( B ∪ C )- A , we will show that ( B- A ) ∪ ( C- A ) ⊆ ( B ∪ C )- A and that ( B- A ) ∪ ( C- A ) ⊇ ( B ∪ C )- A . First, we will show that ( B- A ) ∪ ( C- A ) ⊆ ( B ∪ C )- A . Suppose that x ∈ ( B- A ) ∪ ( C- A ) . By the definitions of difference and union, (( x ∈ B ) ∧ ( x / ∈ A )) ∨ (( x ∈ C ) ∧ ( x / ∈ A )) is true. From Distributive laws (from logic), it follows that (( x ∈ B ) ∨ ( x ∈ C )) ∧...
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This note was uploaded on 02/12/2012 for the course EECS 203 taught by Professor Yaoyunshi during the Spring '07 term at University of Michigan.

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Homework 3 - EECS 203 Homework 3 Solutions Section 2.1 1(E 8defg d True e True f True g False since{∅{∅ ={∅ 2(E 22 a No since a power set

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