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Finding Counterexamples

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

Finding counterexamples requires checking the exact wording of the claim. The best counterexample meets the hypothesis and breaks the conclusion. Learning objective: Search systematically for cases that make a claim false. 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 (Finding Counterexamples): Worked example: Which rule or habit best matches Finding Counterexamples?

  1. Finding Counterexamples targets a specific reasoning habit.
  2. Finding Counterexamples focuses on search systematically for cases that make a claim false.
  3. The other choices either overclaim or change the logical relationship.

Answer: Finding Counterexamples focuses on search systematically for cases that make a claim false.

Practice problems

1. Practice case A (Finding Counterexamples): 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 (Finding Counterexamples): 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 (Finding Counterexamples): 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 (Finding Counterexamples): "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 (Finding Counterexamples): "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 (Finding Counterexamples): "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 (Finding Counterexamples): "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 (Finding Counterexamples): Lesson focus: Which transfer task would show readiness for Finding Counterexamples?

Choices: Apply search systematically for cases that make a claim false 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 search systematically for cases that make a claim false in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply search systematically for cases that make a claim false in a new sentence or context.

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