Question

For each of the following statements, either prove the statement or give a counterexample that shows the statement is false. We will use the (non-standard) notation I to represent the irrational numbers Each problem is worth 10 points. 1. For all mEN2, m2-1 is composite. 2. For all integers a and b If ab is even then a is even or b is even. 3. For all integers a, b, and c If ale and ble then ablc

Answer 1

Let m = 3 then m^2 - 1 = 8, a composite Hence true for m = 3 Let the result be true for m = k > 3 then m^2 - 1 = k^2 - 1 is composite Now, for m = k + 1 m^2 - 1 = ((k + 1)^2) - 1 = k^2 + 2k = k(k + 2) So, m^2 - 1 is product of k, k + 2 where k > 3 hence if is composite Hence by the method of mathematical induction the result is true (2) If possible let ab is even, and a, b, with odd, a = 2k + 1, b = 2k + 1, u, p elementof z then ab = (2u + 1) (2p + 1) = 4kp + 2k + 2p + 1 = 2n + 1 when n = 2up + k p elementof Z which is odd As have a contradiction to the Fact that ab is even, so. the result is true (3) Let c = 4, a = 2, b = 4 then a/c and b/c but ab = 8/4 = c Hence the result is false