Using the following example and week 5 slides, prove the items below are true: Definition: An integer n is even if there exists an integer k such that n= 2k. An integern is odd if there exists an integer k such that n= 2k+ 1 Claim: Let n be an odd integer. Then n2 is odd. Step 1: Checking some easy cases of the claim, we see that when n=-3,(-3)2= 9 is odd; when n=-1,(-1)2= 1 is odd; when n= 1,12= 1 is odd; Whenn=3,32=9 is odd; Step 2: Our premises state that n is an odd integer, namely that there is an integer k such that n= 2k+ 1. The conclusion, which we want to deduce, is that n2 is odd. What would it take to show that n?is odd? We have to find an integer j such that n2= 2j+ 1. Step 3: How do we find this integer j? We need some way to use our information about n to deduce something about n2. We have an equation for n, so square it! This gives us n2= (2K+ 1)2 = 4k2+ 4k+ 1 = 2(2k2+ 2k) + 1. Since 2k2+ 2k is an integer, setting j= 2k2+ 2k yields n2= 2j+ 1, so n2 is odd. We've finished our scratch work, so here is the proof: Proof. Since n is odd, there exists an integer k such that n= 2k+ 1. Then n2=(2k+ 1) = 4k2+ 4k+ 1 = 2(2k2+ 2k) + 1, so n2 is odd. ASSIGNMENT Let n be an integer. Prove the following claims and specify the type of proof (direct or contradiction): (a) Let n be an even integer. Then n2 is even. (b) Let n be an odd integer. Then n3 is odd. (c) Let n be an integer. If n2 is odd, then n is odd. (d) Let n be an integer. If n2 is even, then n is even