Study Notes
Proof by contradiction is a method where you assume the opposite of what you want to prove and show that this assumption leads to a contradiction.
- Proof by contradiction — a method of proving statements by assuming the opposite and showing it leads to a contradiction. Example: To prove that if p² is even, then p is even, assume p is odd and show this leads to a contradiction.
Exam Tips
Key Definitions to Remember
- Proof by contradiction: Assume the opposite of what you want to prove and show it leads to a contradiction.
Common Confusions
- Assuming the wrong initial statement in proof by contradiction.
- Forgetting to reach a contradiction to complete the proof.
Typical Exam Questions
- Use proof by contradiction to show that there is no greatest positive rational number? Assume there is a greatest positive rational number and show this leads to a contradiction.
- Use proof by contradiction to show that there exist no integers a and b for which 25a + 15b = 1? Assume such integers exist and show this leads to a contradiction.
- Disprove the statement "If m and n are irrational numbers, where m≠n, then mn is also irrational" by means of a counter example? Provide a counter example where m and n are irrational but mn is rational.
What Examiners Usually Test
- Understanding the logical steps in proof by contradiction.
- Ability to identify and construct a valid contradiction.
- Application of proof by contradiction to different types of mathematical statements.