Mathematical Proofs and Induction: Set Theory, Even/Odd Properties,and Sum of CubesJeremy BreitMarch 25, 2013MTH 231Homework #2Problem:Prove that, for all integers n, if n2+ 3 is even, then n is odd.Solution:1. Assuming n2+ 3 is even, we can say that 2 divides the equation. n2+ 32|(𝑛2+3)⟺𝑛2+3=2𝑘𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑘𝜖ℤSimply,𝑛2+3is some integer multiple of 2.2. Since 2(k-2) is divisible by 2, and 2 is not divisible by 1, we can say thefollowing:𝑛2=2𝑘−3=2(𝑘−2)+1⟹2𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑑𝑖𝑣𝑖𝑑𝑒2(𝑘−2)+1,⟹2𝑑𝑜𝑒𝑠𝑛𝑜𝑡𝑑𝑖𝑣𝑖𝑑𝑒𝑛2Simply, since 2 does not divide𝑛2,𝑛2cannot be even. This means that𝑛2is odd.3. Next we need to prove that since𝑛2is odd,𝑛is therefor odd. To do this, wewill prove the contrapositive. In this case, we will assume𝑛is even.𝑛=2𝑘𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑘Therefor,𝑛2=(2𝑘)2=2(2𝑘2)4. Since 2 divides𝑛2,𝑛2must be even. Therefor, is n is not odd,𝑛2is not odd aswell.5. Contrapositively, since𝑛2is odd,𝒏must be odd as well.Preview Mode
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