Key Idea: This topic is about written proofs: induction, contradiction and counterexample. Marks come from the structure, so you show every step. It is mainly on Paper 1.
Proof by induction
- Base case — Prove it for n = 1 (check both sides).
- Assume — Assume it is true for n = k.
- Step — Prove n = k + 1 and point to where you use the assumption.
- Conclude — True for n = 1 and k ⇒ k + 1, so true for all n ∈ ℤ⁺.
Proof by contradiction
- Write 'Assume, for contradiction, that …'.
- Follow the logic until something impossible appears.
- Name what is impossible, reject the assumption, state the result.
Disproving with a counterexample
- One value that breaks a 'for all' claim disproves it.
- Check it fits the claim's conditions but breaks the conclusion.
- Look at 0, 1, negatives, fractions and edge cases first.
Induction proves, contradiction proves, one counterexample disproves.
In the odd-numbers proof, what do you add to k²? The next odd term, 2k + 1, giving (k + 1)².
Which induction step must use the assumption? The step from n = k to n = k + 1.
How do you disprove 'irrational + irrational is irrational'? √2 + (−√2) = 0, which is rational.
First line of a contradiction proof? Assume the opposite of the statement.