Mathematical Processes and Problem Solving
The Mathematical Processes and Problem Solving domain assesses deductive reasoning, proof, representation, and the deliberate use of problem-solving strategies. Questions ask you to justify a claim, identify the logical structure of an argument, or select a valid approach rather than merely obtain a number.
Logic and conditional statements
A conditional statement p → q has a converse (q → p), an inverse (¬p → ¬q), and a contrapositive (¬q → ¬p). Only the contrapositive is logically equivalent to the original. A single counterexample disproves a universal statement, but no finite set of examples proves one.
Proof techniques
Direct proof chains known facts to the conclusion. Proof by contradiction assumes the negation and derives an impossibility. Mathematical induction requires a base case and an inductive step: assume P(k) for an arbitrary k at or beyond the base and prove P(k+1).
Strategies
Working backward from the goal, checking parity, applying the pigeonhole principle, and searching for an invariant all convert unfamiliar problems into familiar ones. Representing the same quantity two ways often exposes the needed equation.
Worked example
Claim: the sum of two odd integers is even. Write the odd integers as 2m + 1 and 2n + 1 with m and n integers. Then (2m+1)+(2n+1) = 2m+2n+2 = 2(m+n+1). Because m+n+1 is an integer, the sum is even. Contrast the claim that n² + n + 41 is prime for every nonnegative integer n: it holds from n = 0 to 39 but fails at n = 40 and n = 41, so one counterexample settles it.
Traps
Confusing the converse with the contrapositive; treating examples as proof of a universal claim; omitting the base case; assuming the statement to be proved inside the inductive step.