Further Proof Techniques

This final lesson surveys the proof techniques required at AQA Further Mathematics level, including proof by contradiction, proof by contrapositive, and consolidation of proof by induction. Being able to construct rigorous proofs is a key skill that distinguishes Further Mathematics.

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Proof by Contradiction

Method: Assume the statement is false (assume the negation). Then show this leads to a logical contradiction. Therefore the original statement must be true.

Worked Example 1: Prove that $\sqrt{2}$2​ is irrational.

Assume $\sqrt{2}$2​ is rational, so $\sqrt{2} = \frac{a}{b}$2​=ba​ where $a, b$a,b are integers with no common factor.

Squaring: $2 = \frac{a^2}{b^2}$2=b2a2​, so $a^2 = 2b^2$a2=2b2.

Therefore $a^2$a2 is even, so $a$a is even. Let $a = 2k$a=2k.

Then $4k^2 = 2b^2$4k2=2b2, so $b^2 = 2k^2$b2=2k2, meaning $b^2$b2 is even and $b$b is even.

Both $a$a and $b$b are even — contradiction (they share no common factor).

Therefore $\sqrt{2}$2​ is irrational. ■

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Proof by Contrapositive

The contrapositive of "if $P$P then $Q$Q" is "if not $Q$Q then not $P$P". These are logically equivalent.

Worked Example 2: Prove that if $n^2$n2 is even, then $n$n is even.

Contrapositive: If $n$n is odd, then $n^2$n2 is odd.