7.5PureStretch

Methods of Proof

Beyond direct deduction there are three more tools every A-Level student needs: disproof by counterexample, proof by exhaustion, and proof by contradiction. Choosing the right one is half the battle.

30 min Video by Zeeshan Zamurred Algebraic Methods

Edexcel AS Level Maths: 7.5 Methods of ProofWatch the full walkthrough before the notes below.

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What you'll be able to do

Disprove a statement with a single counterexample
Prove a statement by exhaustion (checking all cases)
Prove a statement by contradiction
Choose an appropriate method for a given claim

Disproof by counterexample

To disprove a statement you only need $one$ example where it fails. This is the fastest method — but it disproves, it never proves.

1Try a value between 0 and 1, say

x = \frac{1}{2}

(\frac{1}{2})^{2} = \frac{1}{4}

, which is less than

\frac{1}{2}

Answercounterexample

x = \frac{1}{2}

disproves it

Tip — A single counterexample is enough to kill a “for all” statement — look at fractions, 0, 1 and negatives.

Proof by exhaustion

When there are only finitely many cases, you can prove a statement by checking $every$ one. Split the problem into a complete set of cases and verify each.

1The primes in range are

31

and

37

2Both are odd, and these are the only cases.

Answertrue by exhaustion

Proof by contradiction

Assume the $opposite$ of what you want to prove, then reason until you reach an impossibility. The contradiction shows the assumption was false, so the original statement is true. This is how $2$ is shown to be irrational.

Assume \neg P \Rightarrow contradiction \Rightarrow P is true

Assume the negation, derive nonsense, conclude the original.

Formula recap

1 counterexample \Rightarrow statement false

Disproof.

check all cases \Rightarrow proof by exhaustion

Finite cases only.

assume \neg P \to contradiction

Proof by contradiction.

Common mistakes to avoid

Trying to prove a “for all” statement by giving one example.

One example proves nothing general; one counterexample only disproves.

Using exhaustion when there are infinitely many cases.

Exhaustion needs a finite, complete set of cases.

Key takeaways

Counterexample: one failing case disproves a statement.
Exhaustion: check every case (only works when finitely many).
Contradiction: assume the negation, reach an impossibility, conclude the statement.
Pick the method that fits the structure of the claim.

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