Reasoning and Mathematical Communication
This competency is assessed inside every other question: a correct number with no valid justification earns little. The exam looks for a clear chain from hypotheses to conclusion, correct notation, and honest handling of evidence.
Valid versus invalid reasoning
Inductive reasoning builds a conjecture from examples; it suggests but never proves. Deductive reasoning moves from definitions and established properties to a necessary conclusion. To prove a universal statement, argue in general (use an arbitrary n); to disprove it, one counterexample is enough.
Tools you are expected to use
- Counterexample: "all prime numbers are odd" fails because 2 is prime and even.
- Contrapositive: to prove "if P then Q", prove "if not Q then not P".
- Converse: "if Q then P" is not equivalent to the original statement.
- Proof by cases or exhaustion: every integer is even or odd, so a property can be checked for both forms.
- Notation and units: =, ≈, ≠, ⇒, ⇔ used correctly; hypotheses stated before they are used.
Worked example
Prove: if n² is even, then n is even. Proof by contrapositive. Assume n is odd, so n = 2k + 1 for some integer k. Then n² = (2k + 1)² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which has the form 2m + 1 and is therefore odd. So whenever n is odd, n² is odd; equivalently, if n² is even, then n must be even. The argument is general - no numerical example was used as proof, which is exactly what the marking rubric rewards.
Traps that cost marks
- Checking three cases and presenting them as a proof.
- Offering a counterexample as proof that a statement is true (it only proves it false).
- Proving the converse instead of the statement, or assuming the conclusion.
- Skipping the hypothesis ("let n be an integer") or mixing ≈ and =.
- Describing a graph or table loosely instead of reading the specific values the question asks about.