Practice question · Sort into groups
Sort each opening line by the proof technique it belongs to.
Groups: Direct proof · Contrapositive · Contradiction
- Assume q fails; we show p must fail too
- Assume there is no largest prime is false, i.e. assume finitely many primes
- Suppose, for the sake of argument, that the statement is false
- Assume p holds; we show q follows
Hints
- Contradiction assumes the whole statement is false; contrapositive assumes only the conclusion fails.
- Both indirect methods start from a negation, but they negate different things.
Show the answer
Direct proof: Assume p holds; we show q follows
Contrapositive: Assume q fails; we show p must fail too
Contradiction: Suppose, for the sake of argument, that the statement is false, Assume there is no largest prime is false, i.e. assume finitely many primes
Why
A direct proof assumes the hypothesis. The contrapositive assumes the conclusion fails and derives that the hypothesis fails. Contradiction assumes the entire statement is false and hunts for an impossibility, the distinction between o2 and o3 is subtle but real.
Practise Proof by Contradiction and Contrapositive
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