CSCI 2400/SENG 1010
Solutions for practice questions set 1019

1. Example. Integer n is even if there exists an integer k where n = 2k. Using a direct proof, show that if n is even then −n is also even.
   
   

   Let n be an arbitrary integer and suppose that n is even. By the definition of an even number, there exists a value k where n = 2k. So −n = −2k = 2(−k). Clearly, −n is even.

2. Suppose that the domain of discourse is the set of integers. Using a direct proof, prove that, if k+n is even and n+m is even then k+m is even.

   1. Suppose that k + n is even and n + m is even.
   2. Select integers x and y such that k+n = 2x and n + m = 2y.
   3. Adding both sides of those equations, k + n + n + m = 2x + 2y.
   4. So k + m = 2x + 2y − 2n = 2(x + y − n).

3. Integer n is odd if there exists an integer k where n = 2k+1. Using a direct proof, show that if m and n are both odd then m+n is even.

   Suppose m and n are both odd. Select integers x and y so that m = 2x+1 and n = 2y+1. Then m + n = 2x + 1 + 2y + 1 = 2(x + y + 1). So m + n is even.
   
   

4. A real number is rational if it is the ratio of two integers. Prove or disprove that if x and y are rational numbers then xy is a rational number.

   Suppose x and y are rational. Select integers a, b, c and d so that x = a/b and y = c/d. Then xy = (a/b)(c/d) = (ac)/(bd). So xy is rational.