6. Use the argument in the proof of Theorem 2.1.4 to show that there does not exist a rational number s such that s2 = 6.

Consider the following proof that if x and y are two even integers, then their sum is even. Proof: Assume that x and y are two even integers but their sum is not even, then x + y is odd. By definition, there are some integers p, q, and k such that x = 2p, y = 2q, and x + y = 2k + 1. Therefore, 2k + 1 = x + y = 2p + 2q. Rewrite above equation, we have 2(p + q - k) = 1, which is impossible. Which proof method is used in above proof. O Proof by counterexample Direct proof Proof by contraposition O Proof by contradiction

By contradiction

1. For any real number r and any integer r #k, show that a) ()=(¹), and b) () = (-1)k (r+k-1). Hint: use the definition of () from Page 137. r-k