Core proof track
Proof by Contradiction & Contrapositive
Work through these bite-sized lessons. Each subsection has reading, notebook prompts, and an optional AI critique to refine your understanding.
Recommended: move in order, but you can jump around if you’re reviewing specific ideas.
Subsections
Click into a subsection to read, practice, and get feedback on your explanation.
Why Indirect Proof Exists
Indirect proofs reframe a hard proof into an easier one by proving an equivalent or stronger contradiction-based statement.
Contrapositive: The Cleanest Technique
Contrapositives replace "If P then Q" with the equivalent but often much easier statement "If not Q then not P."
Proof by Contradiction: Strategy & Examples
In contradiction proofs, you assume the opposite of what you want to prove and show that assumption leads to an impossibility.
Classic Examples (Irrationality of √2)
Classic contradiction proofs often assume a minimal or reduced form and then show that assumption violates its own conditions.
When NOT to Use Contradiction
Use contradiction only when it simplifies the argument — not when it complicates it unnecessarily.
Practice: Rewrite & Convert Proofs
Rewriting proofs in different forms reveals logical structure and improves versatility in mathematical problem solving.