Proof / Contradiction And Counterexample
Least You Need to Know: Contradiction and Counterexample
Use a counterexample to disprove a universal claim quickly. Use contradiction when assuming the opposite of a claim leads to something impossible.
Least you need to know
- One counterexample is enough to disprove a universal statement.
- A proof by contradiction assumes the negation of the target claim.
- A contradiction is something impossible, such as saying an integer is both even and odd.
- Supporting examples never replace a proof.
- Counterexample and contradiction solve different jobs.
Key notation
- ¬P — not P
- ∀ — for all
- ∃ — there exists
Worked example
- Claim: All prime numbers are odd.\n- Counterexample: 2 is prime and even, so the claim is false.\n- This needed only one example because the claim said all.
Common mistakes
- Students often give many confirming examples instead of one disconfirming example.
- Students often assume the converse instead of the negation when using contradiction.
- Students sometimes think contradiction and counterexample are interchangeable.
How to recognize it
- If the claim says all or every and you suspect it is false, look for one counterexample.
- If you need to prove a statement and a direct route is awkward, contradiction may help.
- In contradiction, write down clearly what you are assuming.
Next recommended lesson
Continue through this topic with Least You Need to Know: Contradiction Patterns.
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