CMClearMathAcademy

What a Counterexample Does

A free Logic lesson from the “Counterexamples, Sets, and Diagrams” unit, with a worked example and practice problems including step-by-step solutions.

A counterexample is a specific case that satisfies the setup but fails the conclusion. It is one of the fastest ways to test an all claim. Learning objective: Explain how one example can disprove a universal claim. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: One odd number that is not prime, such as 9, disproves 'All odd numbers are prime.' Example 2: If all squares are rectangles, the square set belongs inside the rectangle set. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.

What you'll learn

Why it matters: Set diagrams help organize categories in math, science, data analysis, and classification tasks.

Worked example

Problem. Example case A (What a Counterexample Does): Worked example: Which rule or habit best matches What a Counterexample Does?

  1. What a Counterexample Does targets a specific reasoning habit.
  2. What a Counterexample Does focuses on explain how one example can disprove a universal claim.
  3. The other choices either overclaim or change the logical relationship.

Answer: What a Counterexample Does focuses on explain how one example can disprove a universal claim.

Practice problems

1. Practice case A (What a Counterexample Does): Which number is a counterexample to "All odd numbers are prime"?

Choices: 9 · 3 · 5 · 7

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (What a Counterexample Does): Which number is a counterexample to "All odd numbers are prime"?
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. A counterexample must be odd and not prime.
  4. 9 is odd.
  5. 9 is not prime, so it disproves the claim.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: 9

2. Practice case B (What a Counterexample Does): If all squares are rectangles, what should a Venn diagram show?

Choices: The square circle inside the rectangle circle · The rectangle circle inside the square circle · No overlap · Only one circle

Show solution
  1. Every square belongs to the rectangle set.
  2. That means the square set is contained in the rectangle set.
  3. The reverse is not guaranteed.

Answer: The square circle inside the rectangle circle

3. Practice case C (What a Counterexample Does): "Some A are B" means:

Choices: At least one object is in both A and B · Every A is B · No A is B · Every B is A

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case C (What a Counterexample Does): "Some A are B" means:
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. Some means at least one.
  4. For set diagrams, that object lies in the overlap.
  5. It does not mean all.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: At least one object is in both A and B

4. Practice case D (What a Counterexample Does): "No A are B" means:

Choices: The sets do not overlap · A is inside B · B is inside A · At least one object is in both

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case D (What a Counterexample Does): "No A are B" means:
  2. For a counterexample, find one case that satisfies the setup but makes the conclusion false.
  3. No A are B rules out shared members.
  4. A Venn diagram would show disjoint sets.
  5. So there is no overlap.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: The sets do not overlap

5. Practice case E (What a Counterexample Does): Lesson focus: Which transfer task would show readiness for What a Counterexample Does?

Choices: Apply explain how one example can disprove a universal claim in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.

Show solution
  1. A transfer task keeps the same objective but changes the surface context.
  2. Apply explain how one example can disprove a universal claim in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply explain how one example can disprove a universal claim in a new sentence or context.

Practice this interactively with instant feedback and an AI tutor.

Practice What a Counterexample Does Take the free placement check

More Logic lessons