Download Prove that if n is an integer and 3n +2 is even, then n

1.6 exercise 18 This proof by contraposition and proof by contradiction are very similar. For proof by contraposition of a statement p  q, assume ~q and show ~p. For proof by contradiction assume ~(pq), which is equivalent to assuming (p ^ ~q), and show that it leads to a contradiction. Prove that if n is an integer and 3n +2 is even, then n is even using a a) proof by contraposition Prove the statement by proving the contrapositive. Assume n is odd and show that 3n + 2 is odd. n = 2k +1 for some integer k by the definition of odd integers. 3n +2 = 3(2k+1) + 2 = 6k +5 = 6k +4 +1 =2(3k+2) + 1, so 3n+2 is 2 times an integer + 1, so it is odd. b) proof by contradiction: Show that if the statement is F, it leads to a contradiction. The statement is F when 3n + 2 is even, and n is odd, so assume 3n + 2 is even, and n is odd. n = 2k +1 for some integer k by the definition of odd integers. 3n +2 = 3(2k+1) + 2 = 6k +3 +2 = 6k +4 +1 =2(3k+2) + 1, so 3n+2 is odd, but we assumed 3n+2 was even. The contradiction completes the proof. There were questions in class about exercise 11 and 12 in 1.7 that we did not get to. I think the question about #12 was based on a mis‐interpretation of the question. 1.7 exercise 12 (homework problem) Prove or disprove that if a and b are rational numbers then ab is also rational. Solution: Example 10 in 1.6 shows that sqrt(2) (which is equivalent to 21/2) is irrational. Let a = 2 and let b = ½, then ab is irrational. This disproves the statement with a counterexample. 1.7 Exercise 11. This was assigned as a review problem. This is a fairly tricky proof, and is the kind of thing you should expect as an extra credit problem. Prove or disprove that there is a rational number x and an irrational number y such that xy is irrational. The solution is given in the back of the book, but it might not be clear why this solution is valid, or how you should come up with it. Begin the proof by following Example 11 on p91. When you get to the second case (in which 2^(21/2) is rational), apply backwards reasoning to understand why a solution is to let y = sqrt(2)/4. That value was chosen because it is the value that sets xy = sqrt(2), and we already know sqrt(2) is irrational. If you don’t master this proof, don’t be concerned—if it, or something like it, appears on the test, it will be for extra credit.