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.
All A are B means inclusion A⊆B; some A are B means a nonempty intersection; no A are B means an empty intersection. Do not reverse inclusion. For a doubtful inference, construct small finite sets satisfying its premises and test the conclusion.
What you'll learn
- Use set language to describe categories and membership
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Let A={28, 29} and B={28, 29, 30}. Which is true?
- Each member of A is in B.
- B contains one additional element.
Answer: A is a proper subset of B
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. Let A={13, 14} and B={13, 14, 15}. Which is true?
Choices: B is a subset of A · A and B are disjoint · A equals B · A is a proper subset of B
Show solution
- Each member of A is in B.
- B contains one additional element.
Answer: A is a proper subset of B
2. All violinists in a club are musicians. Which inclusion follows?
Choices: The sets are disjoint · The sets must be equal · The violinist set lies inside the musician set · The musician set lies inside the violinist set
Show solution
- Every violinist belongs to the musician set, but additional musicians are possible.
Answer: The violinist set lies inside the musician set
3. Some cyclists are runners. What must a Venn diagram show?
Choices: Equal sets · At least one member in the overlap · The entire cyclist set inside the runner set · No overlap
Show solution
- Some guarantees one shared member, not complete inclusion.
Answer: At least one member in the overlap
4. No squares are circles. Which set relation expresses this?
Choices: The intersection of the two sets is empty · All squares are circles · The sets must both be empty · Their union is empty
Show solution
- No shared members means an empty intersection.
- Each set can still contain many separate objects.
Answer: The intersection of the two sets is empty
5. All A are B and all B are C. What follows?
Choices: All C are A · No A are C · Some C must lie outside B · All A are C
Show solution
- An A belongs to B, so it also belongs to C.
- Inclusion is transitive, but cannot be reversed.
Answer: All A are C
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