Midterm II Solutions

Midterm II Solutions - MATH 74 MIDTERM 2 All theorem...

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MATH 74 MIDTERM 2 All theorem references refer to the numbers on the note sheet. 1. A = { 1 } and B = { 2 } will work. [More generally, any two sets with the property that neither is a subset of the other will work] 2. Z = { (1 , 1) , (2 , 2) , (3 , 3) } will work. [Other examples possible; this is related to problem 5 on Homework 6.] 3. The function n 7→ 2 n will work. [Many other examples possible] 4. The function x 7→ x 3 - x will work. [Many other examples possible] 5. A = { 1 } , B = { 2 } , C = { 1 , 2 } will work. [Many other examples possible] I. We will show that #( P ( X )) = #( P ( Y )) by exhibiting a bijection between P ( X ) and P ( Y ) and applying Theorem 6. Define F : P ( X ) P ( Y ) by F ( S ) = { f ( s ) : s S } , S P ( X ) . We will show that F is a bijection. Since f is a bijection, it is invertible by Theorem 1. There is thus a function g : Y X satisfying g ( f ( x )) = x and f ( g ( y )) = y for all x X and y Y . Define G : P ( Y ) P ( X ) by G ( T ) = { g ( t ) : t T } , T P ( X ) . Then for any S P ( X ) we have G ( F ( S )) = G ( { f ( s ) : s S } ) = { g ( f ( s )) : s S } = { s : s S } = S and for any T P ( Y ) we have F ( G ( T )) = F ( { g ( t ) : t T } ) = { f ( g ( t )) : t T } = { t :
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