CMClearMathAcademy

Sets and Categories

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

Sets organize objects by membership. Statements about all, some, and no can often be pictured as relationships between sets. Learning objective: Use set language to describe categories and membership. 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 (Sets and Categories): Which number is a counterexample to "All odd numbers are prime"?

  1. Worked Example: First identify exactly what the question is asking: Example case A (Sets and Categories): Which number is a counterexample to "All odd numbers are prime"?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. A counterexample must be odd and not prime.
  4. 9 is odd.

Answer: 9

Practice problems

1. Practice case A (Sets and Categories): 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 (Sets and Categories): Which number is a counterexample to "All odd numbers are prime"?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  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 satisfies the original prompt and that the other choices fail the same test.

Answer: 9

2. Practice case B (Sets and Categories): 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 (Sets and Categories): "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 (Sets and Categories): "Some A are B" means:
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  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 satisfies the original prompt and that the other choices fail the same test.

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

4. Practice case D (Sets and Categories): "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 (Sets and Categories): "No A are B" means:
  2. For data questions, identify what each statistic measures before calculating so the result matches the question.
  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 satisfies the original prompt and that the other choices fail the same test.

Answer: The sets do not overlap

5. Practice case E (Sets and Categories): Which claim can be disproved by one counterexample?

Choices: All prime numbers are odd. · Some prime numbers are odd. · There exists an even prime. · 2 is prime.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Practice case E (Sets and Categories): Which claim can be disproved by one counterexample?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Universal claims can be disproved by one counterexample.
  4. 2 is prime and not odd.
  5. That breaks all prime numbers are odd.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: All prime numbers are odd.

Practice this interactively with instant feedback and an AI tutor.

Practice Sets and Categories Take the free placement check

More Logic lessons