Venn Diagrams for "No"
A free Logic lesson from the “Counterexamples, Sets, and Diagrams” unit, with a worked example and practice problems including step-by-step solutions.
No A are B means the categories do not overlap. It is a strong universal negative claim. Learning objective: Represent no statements with disjoint sets. 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
- Represent no statements with disjoint sets
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Venn Diagrams for "No"): Worked example: What does "No A are B" mean in a set diagram?
- No A are B rules out any shared members.
- So the circles are drawn disjoint.
- There is no overlap region.
Answer: The A and B circles do not overlap.
Lesson preview: this page shows 5 practice items. The interactive activity contains 20 practice items and 10 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. Practice case A (Venn Diagrams for "No"): Practice: What does "No A are B" mean in a set diagram?
Choices: The A and B circles do not overlap. · A is inside B. · B is inside A. · At least one object is in both.
Show solution
- No A are B rules out any shared members.
- So the circles are drawn disjoint.
- There is no overlap region.
Answer: The A and B circles do not overlap.
2. Practice case B (Venn Diagrams for "No"): Practice: How is "no overlap" different from "not all A are B"?
Choices: No overlap means zero shared members; 'not all' just means at least one A is outside B. · They mean the same thing. · 'Not all' means the circles are disjoint. · 'No overlap' allows some shared members.
Show solution
- 'No A are B' = disjoint circles.
- 'Not all A are B' still allows many A to be B.
- The two claims have very different strength.
Answer: No overlap means zero shared members; 'not all' just means at least one A is outside B.
3. Practice case C (Venn Diagrams for "No"): Practice: A diagram shows circle A entirely inside circle B. Which statement matches?
Choices: All A are B. · Some A are B (and some are not). · No A are B. · All B are A.
Show solution
- A inside B means every A is a B.
- That is the 'All A are B' picture.
- It does not show that all B are A.
Answer: All A are B.
4. Practice case D (Venn Diagrams for "No"): Practice: A diagram shows circles A and B overlapping, with one dot in the overlap. Does this prove "All A are B"?
Choices: No — it only shows some A are B; parts of A may be outside B. · Yes — any overlap proves all A are B. · No — it proves no A are B. · Yes — one dot is enough for all.
Show solution
- One shared member shows 'some,' not 'all.'
- Region of A outside B can still exist.
- So the diagram leaves 'all A are B' unproven.
Answer: No — it only shows some A are B; parts of A may be outside B.
5. Practice case E (Venn Diagrams for "No"): Practice: In a Venn diagram, "All A are B" looks like:
Choices: Circle A drawn completely inside circle B. · Circle B drawn completely inside circle A. · Two circles that do not touch. · Two circles that partly overlap.
Show solution
- All A are B means every A is also a B.
- So A sits entirely within B.
- It does not force every B to be an A.
Answer: Circle A drawn completely inside circle B.
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