ICS 6B
Fall 2013
Homework 3


Due: Wednesday, Oct 23

Covers Sections 1.8

In all of the proofs below, state what method of proof you are using. At the beginning of each proof, state what you are assuming and what the proof will show.

  1. Prove that if n is an odd integer, then n2 is an odd integer.

  2. Prove that if n2 is an odd integer, then n is an odd integer.

  3. Prove that if a2 - 2a + 7 is even then a is odd.

  4. Prove that if a is an integer and a2 is a multiple of 3, then a2 is a multiple of 9.

  5. Prove that √ 3  is irrational.

  6. Prove that if a and b are positive real numbers, then a + b ≥ √ ab  .

  7. Prove that if x is a real number in the range 0 ≤ x ≤ 3, then 12 -7x + x2 ≥ 0.

  8. Prove that if x and y are real numbers, then max(x, y) + min(x, y) = x + y. (Hint: do a proof by cases with the two cases being x ≤ y and x > y.)

  9. Prove that if x is an irrational number that is not equal to 0, then 1/x is also an irrational number.

  10. Prove that if an integer n is a perfect square, then n + 2 is not a perfect square.