Problem Solving, Modelling and Proof
Problem Solving, Modelling and Proof tests how you construct arguments, choose representations and validate models. Expect direct proof, contradiction, contrapositive, induction, counterexamples, and modelling cycles with assumptions.
Core rules
- Direct proof: start from definitions and known truths, then derive the conclusion.
- Contradiction: assume the negation and derive an impossible statement. Contrapositive proves ¬Q⇒¬P instead of P⇒Q.
- A counterexample disproves a universal statement; one example cannot prove it.
- Modelling: define variables and units, state assumptions, build equations, interpret results, and validate against context.
- Induction is appropriate for statements indexed by integers; proof by exhaustion is valid only for a finite exhaustive list.
Worked example
Prove that no integer n has n²≡2 mod 4. Any integer is even or odd. If n=2k, then n²=4k²≡0 mod 4. If n=2k+1, then n²=4k²+4k+1≡1 mod 4. Since squares modulo 4 are only 0 or 1, the residue 2 is impossible.
Traps
Proving the converse; using an example as proof; ignoring domain restrictions; leaving assumptions unstated; failing to check a model against data; treating a necessary condition as sufficient.