Building Mathematical Arguments

Discrete Math · Axiom Academy

LESSON Building Mathematical Arguments Master the four fundamental proof strategies: direct proof, contradiction, contrapositive, and proof by cases—with concrete examples from number theory. 1. Direct Proof: The Straightforward Path When to use: When there's a clear path from hypothesis to conclusion. Direct proof is your first choice—try it before considering other methods. 2. Proof by Contradiction: Assume the Opposite When to use: When the statement involves negation ("no such thing exists," "infinitely many," etc.) or when direct proof seems blocked. 3. Proof by Contrapositive: Reverse the Logic When to use: When the hypothesis is hard to work with but the negation of the conclusion gives you something concrete to start from. 4. Proof by Cases: Divide and Conquer When to use: When the hypothesis naturally splits into distinct scenarios, or when definitions involve "or" statements. Now that you've seen all four methods, here's how to choose which one to use: Direct Proof: Try this first! If you can see a clear path from hypothesis to conclusion. Contrapositive: When the negation of the conclusion is easier to work with than the hypothesis. Contradiction: When proving something doesn't exist, or when both direct and contrapositive seem blocked. Cases: When the problem naturally splits into scenarios, or involves "or" conditions.

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