216hw3evensol - Name: xxxxxxxxxxxxxxxxxxxxxxxxx Math 216...

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Unformatted text preview: Name: xxxxxxxxxxxxxxxxxxxxxxxxx Math 216 – Discrete Structures (2) Homework: Logic II 50 1. (10) (1.6: 6) Use a direct proof to show that the product of two odd integers is odd. Let a,b be odd integers. Then we can write a = 2 x + 1 and b = 2 y + 1 for some integers x,y . Multiplying a and b gives: ab = (2 x + 1)(2 y + 1) = 4 xy + 2( x + y ) + 1 = 2(2 xy + x + y ) + 1 Therefore the product of two odd integers is odd. 2. (1.6: 12) Prove or disprove that the product of a nonzero rational number and an irrational number is irrational. By contradiction: Let x ∈ Q \ { } and y ∈ R \ Q . Assume that xy = z ∈ Q . Then there exists a,b,c,d ∈ Z such that x = a b , z = c d Then xy = z y = zx- 1 y = cb ad ∴ y ∈ Q Since y ∈ R \ Q , this contradicts our assumption and the product of a nonzero rational number and and irrational number is irrational. 3. (1.6: 16) Prove that if m and n are integers and mn is even then m is even or n is even. 1 Name: xxxxxxxxxxxxxxxxxxxxxxxxx Proof by contradiction: Assume both m and n are odd and mn is even. Then from the definition of odd integers there exists k 1 ,k 2 ∈ Z such that m = 2 k 1 +1 and n = 2 k 2 + 1 . Then....
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216hw3evensol - Name: xxxxxxxxxxxxxxxxxxxxxxxxx Math 216...

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