how to do proofs with algebraic propertys

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Even An integer n is defined to be even when n=2k for some integer k. Odd An integer n is defined to be odd when n=2k +1 for some integer k. Application Now we will look at a basic proof in abstract mathematics. If x is even then x2 is even. Notice that the statement above has an assumption and a conclusion. The assumption is that x is even while our conclusion is that x2 is even. Now here is the formal proof. Proof: Assume that x is even. [We wish to show that x2 is even] By definition of even x=2k for some integer k. Therefore, x2= (2k) 2= 4k2. Moreover, x2 =4k2=2(2k2). Since k is an integer then 2k2 is an integer. Hence, we conclude that x2 is even.