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.
The 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
Tiny 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 this kind of problem
- 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.