Question

Question 3 ONLY

Notes: This homework consists of 3 problems. Write your answers in full sentences. Before you start a problem set up the goal and explain how will you proceed and why. Write all details thinking that the grader knows nothing about Mathematics. Pictures and graphs are great ideas to start a proof but will not be sufficient without written explanation. If you are referring to some result or theorem, clearly write its statement and location in the text. 1. Let x e Z. Prove that 3x+2 is even if and only if 5x 3 is odd. 2. Let a, b e Z. Prove that a (b1) is even if and only if a is even or b is odd. 3. Prove, for every integer n, that the integers 5n 1 and 7n +2 are of opposite parity.

Answer 1

Without loss of generality assume that 5n + 1 is odd integer ie 5n + 1 = 2m + 1 for some m elementof7L Now 7n + 2 = 5n + 2n + 1 + 1 = (5n + 1) + (2n + 1) = 2m + 1 + 2n + 1 = 2(m + n + 1) = even. Hence 5n + 1 & 7n + 2 are of opposite parity. Now assume that 5n + 1 is even. Rightarrow 5n + 1 = 2m for some melementof7L Now 7n + 2 = 5n + 1 + 2n + 1 = 2m + 2n + 1 = 2(m + n) + 1 = odd Hence proved.