Core proof track
Direct Proofs
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.
How Direct Proofs Work
A direct proof of "If P, then Q" starts by assuming P, logically deduces facts using definitions and known results, and ends by showing Q must follow.
Proving Statements About Even/Odd Numbers
For even/odd proofs, always start from n = 2k or n = 2k + 1, do the algebra, and show the result fits the correct even/odd definition.
Proving Divisibility Statements
Divisibility proofs unwrap "d ∣ n" into n = dk, manipulate that equation, and then rewrap it to show divisibility by another integer.
Proving Inequalities
Inequality proofs use algebra plus basic order rules: preserve direction when multiplying by nonnegative numbers, and often rewrite to something obviously ≥ 0 or ≤ 0.
Techniques for Structuring a Direct Proof
Good direct proofs have a clear opening (assumptions), a logically ordered middle, and an explicit closing that states the conclusion has been reached.
Direct Proof Practice Problems
These problems are pure reps: use the same direct-proof patterns (assume, translate definitions, do algebra, translate back, conclude) until the process feels natural.