Proof by Counterexample
A free Logic lesson from the “Proof Readiness” unit, with a worked example and practice problems including step-by-step solutions.
To disprove a universal implication, choose an object inside the stated domain that satisfies its hypothesis but violates its conclusion. A failed example outside the domain is irrelevant. Many successful examples suggest a claim but do not prove it for an unlimited domain.
What you'll learn
- Disprove all claims with a single correct counterexample
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Which integer disproves “Every multiple of 28 is a multiple of 56”?
- 28 satisfies the hypothesis but is not divisible by 56.
Answer: 28
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Which integer disproves “Every multiple of 13 is a multiple of 26”?
Choices: 26 · 52 · 78 · 13
Show solution
- 13 satisfies the hypothesis but is not divisible by 26.
Answer: 13
2. Which value disproves “Every odd positive integer is prime”?
Choices: 5 · 7 · 9 · 3
Show solution
- 9 is odd, positive and composite because 9=3×3.
Answer: 9
3. Which real x disproves x²=x for every real x?
Choices: Both 0 and 1 · 13 · 0 · 1
Show solution
- 13²=169, which is not 13.
- Both 0 and 1 actually satisfy the equation.
Answer: 13
4. A claim concerns all integers greater than 10. A proposed counterexample is 8. Why is it invalid?
Choices: It does not satisfy the hypothesis · A counterexample must be prime · A counterexample must prove the conclusion · One example can never disprove a universal claim
Show solution
- A counterexample must belong to the stated domain and satisfy the setup.
Answer: It does not satisfy the hypothesis
5. A student checks 100 squares and observes four equal sides. Does this alone prove every square has four equal sides?
Choices: Yes; 100 examples always prove a theorem · No; squares do not have equal sides · Yes; a diagram proves every case · No; use the definition to establish the universal claim
Show solution
- Finite examples do not by themselves prove an unbounded universal claim.
- The definition supplies the general justification here.
Answer: No; use the definition to establish the universal claim
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