Mathematical Proofs And Induction: Set Theory, Even/Odd Properties, And Sum Of Cubes

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Mathematical Proofs and Induction: Set Theory, Even/Odd Properties, and Sum of Cubes Jeremy Breit March 25, 2013 MTH 231 Homework #2 Problem : Prove that, for all integers n, if n 2 + 3 is even, then n is odd. Solution : 1. Assuming n 2 + 3 is even, we can say that 2 divides the equation. n 2 + 3 2 | ( 𝑛 2 + 3 ) ⟺ 𝑛 2 + 3 = 2 𝑘 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑘 𝜖 ℤ Simply, 𝑛 2 + 3 is some integer multiple of 2. 2. Since 2(k - 2) is divisible by 2, and 2 is not divisible by 1, we can say the following: 𝑛 2 = 2 𝑘 − 3 = 2 ( 𝑘 − 2 ) + 1 ⟹ 2 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑑𝑒 2 ( 𝑘 − 2 ) + 1 , ⟹ 2 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑑𝑖𝑣𝑖𝑑𝑒 𝑛 2 Si mply, since 2 does not divide 𝑛 2 , 𝑛 2 cannot be even. This means that 𝑛 2 is odd. 3. Next we need to prove that since 𝑛 2 is odd, 𝑛 is therefor odd. To do this, we will prove the contrapositive. In this case, we will assume 𝑛 is even. 𝑛 = 2 𝑘 𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑘 Therefor, 𝑛 2 = ( 2 𝑘 ) 2 = 2 ( 2 𝑘 2 ) 4. Since 2 divides 𝑛 2 , 𝑛 2 must be even. Therefor, is n is not odd, 𝑛 2 is not odd as well. 5. Contrapositively, since 𝑛 2 is odd, 𝒏 must be odd as well.
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