Direct Proof and Counterexample
Chapter 3 consists of properties of integers, rational numbers and
real numbers and how to use these to determine the truth and
falsity of mathematical statements.
ex. If 6x + 9y = 101, then either x or y is not an integer.
Method of exhaustion:
Definitions and Properties:
1. The sum of two integers is an integer.
The difference of two integers is an integer.
The product of two integers is an integer.
2. An integer, n, is even iff (if and only if) n  2k, k  Z .
3. An integer, n, is odd iff n  2k  1, k  Z .
4. An integer, n, is prime iff n >
1 and
r,s  Z  , if n  rs, then r 1 or s 1.

5. An integer, n, is composite iff

 r,s Z , if n  rs, then r  1 and s 1. .
ex.
Prove: If 6x + 9y = 101, then either x or y is not an integer.
Steps:
1. Write the statements to be proved.
x  D, if P(x), then Q(x).
2. Mark the beginning of the proof with the word “Proof.”
3. Identify variables. Suppose
x  D and P(x) .
4. Write proofs in sentences. (i.e. use =)

ex. Prove:
n  Z , if n is even then n2 is even.
Proof:

ex. Prove:
Proof:

n  Z  and n > 1, n! + 1 is odd.
Disprove by counterexample.
ex. For all integers m, if m > 2, then m2 – 4 is composite.
If you only have
m,n  Q,
 in the statement, you can prove by example.
1 1
 Z
m n

Do: Prove or show a counterexample.
1 1

m,n

Q,
  Z.
1.
m n
2. The negative of an even number is even.

n  R, if n is even,  n is even.
3.
 m,n  Z, if mn  1, then m  n  1 or m  n  1.