The document discusses counterexamples and how they can disprove conjectures or general statements. It provides examples of counterexamples, such as a rectangle with an odd perimeter disproving the conjecture that a rectangle's perimeter can never be odd. It also notes that only one counterexample is needed to prove a conjecture false.

1. Counterexamples

2. Conjecture:
The perimeter of a rectangle can never be an odd
number

3. Counterexample:
­ an example that shows a general statement
to be false.
­ it only takes one counterexample to prove
a conjecture false

4. Conjecture:
Every rectangle is a square.
Counterexample:

5. Conjecture:
If a>b and b=5, then a >= 6
Counterexample:

6. Students with low grade­point averages in high
school are invariably failures in life.

7. Exercise 48
Omit question 9