Homework #3
Proof Techniques
Read section 1.4
Exercises from the text: Page 35: #1,#2,#4 a,d, #5,
Other exercises in thought and logic that you will most certainly find interesting.
For each exercise, determine whether the statement is true or false. If true, prove the
statement. If false, give a counterexample.
1. There are infinitely many primes.
2. If there are several people in a room with some who know one another and others
who do not, then among the people there are at least two who know the same number
of people. Note: knowing one-another is a symmetric, non-reflexive relationship.
3. If a and b are irrational numbers, then a + b is irrational.
4. If a and b are irrational numbers, then a  b is irrational.
5. If a and b are irrational numbers, then ab is irrational
6. Among any n positive integers, there exists at least two whose difference is divisible
by n – 1.
7. A standard chess board has 64 squares (8 x 8). Suppose that we have 32 dominoes
that each cover exactly two squares of the chess board; then the entire board can be
covered with the 32 dominoes. Now, cut the two opposite corners from the chess
board as shown. This mutilated chess board can be covered with 31 dominoes?
8. Given a 2n  2n “chessboard” with exactly one square removed, this mutilated
22 n  1
“chessboard” can be covered by
triominoes of the shape below:
3