Learning to Write Math Proofs with AI

A proof explains why a claim follows from definitions, assumptions and established results. MIT OpenCourseWare’s Mathematics for Computer Science notes distinguish proving existence with a valid example from proving a universal statement for an arbitrary member, and cover contradiction and induction. An AI assistant can suggest a route or rephrase a definition, but a polished paragraph is not a substitute for valid logical steps. First rewrite the claim with its quantifiers: for every object, there exists an object, or under a stated condition something follows. Choose a method that fits the claim. A direct proof starts from assumptions and reaches the conclusion. A contrapositive proof addresses an equivalent implication; a contradiction assumes the negation and derives an impossibility. Induction needs a base case, an induction hypothesis for the chosen index, and a step that reaches the next case. An example can prove existence but usually cannot prove a statement about all integers. A single counterexample can refute a universal statement. Ask the model to name what each line uses rather than letting it say 'obvious' at the crucial step. Check common failure modes. Reversing 'if P then Q' does not automatically prove 'if Q then P.' Assuming the conclusion in a disguised form is circular. A proof that checks only small cases may suggest a pattern but not establish it for all cases. In induction, the step must actually use the hypothesis or otherwise justify the next case. Test the proposed argument with edge cases and attempt to identify its weakest implication. For practice, ask for one hint or for a critique of your own draft before seeing a complete proof. Rewrite the argument in your own words and verify every invoked theorem’s conditions. If a teacher permits AI assistance, disclose it under the course rules. The educational outcome is the ability to construct and audit a new proof, not to hand in a plausible-looking generated one.