Directions: Prove the following propositions by using the kind of proof indicated in each number. Use outline form for numbers 1. and 2. while paragraph form for numbers 3. and 4.

Question

Directions: Prove the following propositions by using the kind of proof indicated in each number. Use outline form for numbers 1. and 2. while paragraph form for numbers 3. and 4.

1. If a is an odd integer, then a² + 3a + 5 is odd. (Direct Proof)

2. Suppose x, y € Z. If x is even, then xy is even (Indirect Contraposition Proof)

3. For all real numbers a and b, if a² = b² , then a = b (Proof by Counter Example)

4. For all integers n. if n³ + 5 is odd then n is even. (Proof by Contradiction)

Answer 1

1. a is odd, so a = 2n + 1 for some integer n. So a² + 3a +5 = (2n+1)²+3(2n+1)+5 = 4n²+10n + 9.

4n² and 10n are even and 9 is odd, so 4n²+10n + 9 must be odd.

2. Suppose x is even and xy is odd. Since x is even there exists an integer n such that x = 2n. Thus

xy = (2n)y = 2(ny). Thus xy is even , contradicting our assumption that xy is odd. Therefore the assumption that xy is odd must be false. Thus xy must be even.

3 (-3)² = 3², but -3 ≠ 3

4. Suppose n³ + 5 is odd and n is odd. If n is odd, there exists integer k such that n = 2k + 1. Then

n³+5 = (2k+1)³+5 = (2k)³+3(2k)²(1)+3(2k)(1)²+(1)³+5=8k³+12k²+6k+1+5 = 2(4k³+6k²+3k+3).

Since k is an integer, 4k³+6k²+3k+3 is also an integer, and so n³+5 is even. Thus the assumption that n³+5 is odd must be false. Thus n³+5 must be even.

Answer 2

1. An odd integer multiplied by an odd integer is odd. Therefore if a is odd, a^2 and 3a are both odd, and 5 is odd. The sume of three odd integers is odd. Proved.
2. If x is even, then we can write x = 2a where a is an integer. Then xy = 2ay = 2 (ay). a and y are integers so ay is an integer, and 2 ay must be even.
3. This is untrue. Assume a^2 = b^2. If a = b this is true, but also if a = -b this is true. So a may be b but doesnt have to be.
4. Suppose n was odd. Then n^3 is the product of three odd numbers so must be odd. 5 is clearly odd. Adding odd numbers gives an even number. So if n was odd then n^3 + 5 must be even. But it needs to be odd, so no can't be odd, it must be even.