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Venn Diagrams for "All"

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 the A set sits inside the B set. The diagram does not imply all B are A. Learning objective: Represent all statements with containment diagrams. 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 (Venn Diagrams for "All"): Worked example: In a Venn diagram, "All A are B" looks like:

  1. Worked Example: First identify exactly what the question is asking: Example case A (Venn Diagrams for "All"): Worked example: In a Venn diagram, "All A are B" looks like:
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. All A are B means every A is also a B.
  4. So A sits entirely within B.

Answer: Circle A drawn completely inside circle B.

Practice problems

1. Practice case A (Venn Diagrams for "All"): 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
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Venn Diagrams for "All"): Practice: In a Venn diagram, "All A are B" looks like:
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. All A are B means every A is also a B.
  4. So A sits entirely within B.
  5. It does not force every B to be an A.
  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: Circle A drawn completely inside circle B.

2. Practice case B (Venn Diagrams for "All"): 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
  1. Warm-up: First identify exactly what the question is asking: Practice case B (Venn Diagrams for "All"): Practice: A diagram shows circle A entirely inside circle B. Which statement matches?
  2. For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
  3. A inside B means every A is a B.
  4. That is the 'All A are B' picture.
  5. It does not show that all B are A.
  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: All A are B.

3. Practice case C (Venn Diagrams for "All"): Practice: How is "some overlap" different from "complete inclusion" of A in B?

Choices: Overlap means at least one shared member; inclusion means every A is in B. · They are the same relationship. · Overlap means every A is in B. · Inclusion means the circles do not touch.

Show solution
  1. 'Some overlap' = a non-empty intersection.
  2. 'Complete inclusion' = A entirely within B.
  3. Inclusion is stronger than mere overlap.

Answer: Overlap means at least one shared member; inclusion means every A is in B.

4. Practice case D (Venn Diagrams for "All"): 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
  1. One shared member shows 'some,' not 'all.'
  2. Region of A outside B can still exist.
  3. 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 "All"): Practice: Which value is a counterexample to "All odd numbers are prime"?

Choices: 9 · 3 · 5 · 7

Show solution
  1. A counterexample is one odd number that is not prime.
  2. 9 is odd but not prime, since 9 = 3 × 3.
  3. One counterexample is enough to disprove an all claim.

Answer: 9

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