Mathematical Reasoning, Explanation, and Proof
What this domain tests
Short written justifications: explaining why a statement is true, spotting the flaw in someone else's reasoning, producing a counterexample, and setting out a two- or three-line proof.
Core rules
- Separate what is given from what must be shown, and write the target at the top.
- Every line needs a reason: a definition, a given fact, or a named theorem. A line without a reason earns nothing.
- Prove general statements with algebra, not examples. Represent even numbers as 2m and odd numbers as 2n+1, with m and n integers.
- To disprove a claim, one counterexample is enough; to prove it, examples never are.
- A statement and its converse are different. "If P then Q" does not license "if Q then P".
Worked example
Prove that the sum of two odd integers is even. Let the integers be 2m+1 and 2n+1 with m, n integers. Sum = 2m+1+2n+1 = 2m+2n+2 = 2(m+n+1). Since m+n+1 is an integer, the sum is twice an integer, hence even. ∎
Traps that cost marks
Assuming the conclusion partway through; treating a accurate diagram as proof; testing three cases and declaring a general result; confusing a theorem with its converse; leaving the final "therefore" sentence unwritten.