CSCI 2400/SENG 1010
Solutions for practice questions set 1021

Write clear, readable proofs.

1. Using a proof by contraposition, prove that if x and y are real numbers where x+y ≥ 2 then x ≥ 1 or y ≥ 1 (or both).

   We prove the contrapositive: If x < 1 and y < 1 then x+y < 2. Add inequalities x < 1 and y < 1, giving x+y < 2.

   
   
   

2. A real number is rational if it is the ratio of two integers. The square root of 2 is irrational. Prove that there exist real numbers x and y where x and y are both irrational but xy is rational.

   Choose x and y both to be the square root of 2. x and y are irrational, but their product xy is 2.

   
   
   

3. Prove that, if x is an irrational real number then 1/x is irrational.

   1. We prove the contrapositive: If 1/x is rational then x is rational.
   2. Assume that 1/x is rational. Then 1/x = a/b for two integers a and b.
   3. Notice that 1/x cannot be 0, since no number has a reciprocal of 0. So a ≠ 0.
   4. x = b/a. So x is rational.
   
   
   

4. Prove that there exists a real number x so that x² < x.

   Choose x = 1/2. Notice that (1/2)² < (1/2).

   
   
   

5. Find a counterexample to the statement that every positive integer is the sum of the squares of three integers.

   7 is not the sum of the squares of three integers. The only squares that could be used to sum to 7 are 0, 1 and 4. You can only use 4 once, since 4+4 > 7. But then the largest number you can get is 4 + 1 + 1 = 6.