Proof
What you must understand
Extension 2 proof questions rest on a small toolkit: direct proof, proof by contradiction, proof by contraposition, disproof by counterexample, and equivalence proofs (proving both directions separately). You must also handle inequalities rigorously - establishing a chain of implications from known truths to the target statement, never assuming the conclusion. Familiarity with the language of quantifiers ("for all", "there exists") matters: negating a statement correctly is often the first step of a contradiction argument.
Traps
- Circular reasoning: starting from the statement to be proved and manipulating until something true appears. Only reversible steps, or a clearly stated forward chain, are valid.
- Confusing necessary and sufficient: proving one direction of an "if and only if" and claiming the whole result.
- Counterexample misuse: a single example never proves a universal statement, but one counterexample does disprove it.
- Inequality direction: multiplying or squaring both sides without checking signs; squaring is only safe when both sides are known non-negative.
- Hidden division by zero or by an expression that could vanish.
Rule of thumb
To prove (P \Rightarrow Q) by contraposition, assume (\lnot Q) and derive (\lnot P); this is often far easier than a direct attack when (Q) is a negative or "irrational" statement.
Worked example
Prove that if (n^2) is even then (n) is even, for integer (n).
Contrapositive: suppose (n) is odd, so (n = 2k+1). Then (n^2 = 4k^2+4k+1 = 2(2k^2+2k)+1), which is odd. Hence (n^2) odd implies (n) odd, so (n^2) even implies (n) even.