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# quiz2 - m 2 = n 2 so we conclude this direction is true ∴...

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Name: Math 55 Quiz 2 SOLUTIONS June 26, 2009 GSI: Rob Bayer You have until 4:00 to complete this quiz. You must show your work. 1. (3 pts) Re-write the statement ¬∀ x y ( P ( x ) ∧¬ Q ( y )) so that negations appear only directly in front of predicates. x y ( ¬ P ( x ) Q ( y )) 2. (3 pts) Prove that m 2 = n 2 if and only if m = n or m = - n We’ll do this by showing both directions individually: ( ) Suppose m 2 = n 2 . Then m 2 - n 2 = 0, so ( m - n )( m + n ) = 0. Thus, either m = n or m = - n ( ) Suppose m = n or m = - n . Then if m = n , squaring both sides gives m 2 = n 2 . If m = - n , then squaring both sides gives m 2 = ( - n ) 2 = ( - 1) 2 n 2 = n 2 . In either case,
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Unformatted text preview: m 2 = n 2 , so we conclude this direction is true. ∴ m 2 = n 2 ⇔ m = n or m =-n 3. (4 pts) Prove that if x + y is irrational, then either x is irrational or y is irrational. We’ll do this by contrapositive. That is, we’ll show that x ∈ Q and y ∈ Q ⇒ x + y ∈ Q Suppose x,y ∈ Q and let a,b,c,d ∈ Z such that x = a b ,y = c d . Then x + y = a b + c d = ad + cb bd ∈ Q (since ad + cb,bd ∈ Z ) Thus, we have the desired result....
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