Mathematical Methods and Proof
Core concepts
A candidate must command algebraic manipulation, function behaviour, and the logic of proof. Know the difference between necessary and sufficient conditions, how to negate quantified statements, and the standard proof forms: direct, contrapositive, contradiction, and mathematical induction. Induction requires a clearly stated base case, an explicit inductive hypothesis, and a step that genuinely uses that hypothesis.
Traps
- Confusing the converse with the contrapositive; only the contrapositive is logically equivalent to the original.
- Induction steps that assert the result for n+1 without invoking the hypothesis for n.
- Losing domain restrictions when dividing, squaring, or taking roots; squaring can introduce extraneous solutions.
- Treating a counterexample as a proof of a general statement rather than a disproof.
Worked example
Prove by induction that 1+3+...+(2n-1)=n^2. Base case n=1: LHS=1, RHS=1. Assume true for n=k, so the sum to k equals k^2. For n=k+1 the sum is k^2+(2k+1)=(k+1)^2, completing the step. Hence true for all positive integers.
Rule of thumb
To disprove a "for all" claim, one counterexample suffices; to prove it, induction or a general argument is required.