Skip to main content
MathidiaMathematics Academy
Proof and mathematical reasoning
Lesson 69 of 75
Challenge

Proof, disproof and counterexample

Objective. Choose a valid proof strategy for different claim types.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Proof, disproof and counterexample teaches choosing the right strategy: a single counterexample disproves a general claim instantly, while proving a claim true requires a general argument covering every case. To disprove 'all prime numbers are odd', simply state that 2 is prime and even, which is one valid counterexample. To prove a claim true, use algebraic generalisation (letters standing for any number) rather than specific numeric checks, since no finite set of examples proves a universal statement. Recognising which type of claim you face, a universal 'always' statement or an existence claim, determines whether you search for a counterexample or build a full proof.

Worked example

Give a counterexample to disprove the claim 'the square of every integer is greater than the integer itself'.

Diagram of a square
  1. 1Consider the claim for integer n = 0
  2. 2Square of 0 is 0² = 0
  3. 30 is not greater than 0 itself
  4. 4So n = 0 is a counterexample, disproving the claim

Answer: n = 0 is a counterexample since 0² = 0

Concept mastery

Based on your past attempts at this lesson — the question types to practise again come first.

Complete a practice set to see your concept breakdown here.

Practice

Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.

Start practising

Common mistake. Students often try to disprove a claim using several supporting examples instead of realising that just one well-chosen counterexample is sufficient and that supporting examples never serve as disproof.

Your progress

Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.

0 of 5 steps complete

Your place saves automatically as you read.

My dashboard