Indirect Proof Reference

Intro to Proofs · Axiom Academy

FORMULA SHEET Indirect Proof Methods Reference for contrapositive and contradiction proof techniques Negate both hypothesis and conclusion. Reverse the direction of implication. For quantifiers: keep , negate the inner conditional as above. The negation of q is easier to work with than p . q describes what something is not . Proving divisibility or parity properties. The conclusion involves "or" statements. If ab is even, then a or b is even. If f(a) = f(b) then uniqueness holds. Proof by Contrapositive — Template Proof: We prove the contrapositive . Assume: — state the negation of the conclusion clearly. Work forward from using definitions, properties, and prior results. Apply logical steps and algebraic manipulations. Conclusion: By contrapositive, . Contradiction Method Structure Assume (negation of what you want to prove). Use given information + assumption. Conclude the assumption was false, so P is true. Proof by Contradiction — Template Proof: Assume, for the sake of contradiction, that . List any given conditions or premises. Combine the assumption with the given information. Apply logical reasoning and algebraic manipulation. But this is a contradiction — state explicitly what contradicts what. Therefore: our assumption must be false, so P is true. If a is irrational, then is irrational. There are infinitely many primes. Pigeonhole principle applications.

This is the written version of the interactive lesson above. See the full Intro to Proofs course.