Venn
Draw and read Venn diagrams for two-set logic.
Venn
Venn Diagrams for Two Sets
What you'll learn
- To draw and read two-set Venn diagrams — overlapping circles for two groups.
- Regions: only A, only B, both A and B, neither (outside both).
- To match All / Some / No statements to the correct diagram shading.
- Visual tool for Class 5 syllogism and set reasoning.
NCERT / CBSE link
Math-Magic 5, Chapter 12 (Smart Charts) and CBSE logical reasoning use overlapping categories — Venn diagrams appear in olympiad prep from Class 5 onward.
Key concepts
Level 1 — Four regions
Verbal: Two overlapping circles inside a rectangle (universal set).
Symbolic: n(A only), n(B only), n(A∩B), n(neither).
| Region | Meaning |
|---|---|
| Left crescent | In A, not in B |
| Lens overlap | In both A and B |
| Right crescent | In B, not in A |
| Outside circles | In neither |
Verbal: Shading or ✓ marks the region described by the statement.
Level 2 — Statement → diagram
| Statement | Diagram action |
|---|---|
| All A are B | A circle inside B (or A region ⊆ B) |
| Some A are B | Mark overlap non-empty |
| No A are B | Circles separate, no overlap |
| Some A are not B | Part of A outside B shaded |
Real-life: Students who play cricket AND chess — overlap region.
Worked example
40 students: 25 like math, 20 like science, 10 like both. How many like only math?
Step 1 — Both = 10 → only math = 25 − 10 = 15
Step 2 — Only science = 20 − 10 = 10
Step 3 — Venn: fill overlap first, then crescents.
Answer: 15 like only math.
Draw diagram for: No cats are dogs.
Step 1 — Two separate circles labelled Cats, Dogs.
Step 2 — No overlap region.
Answer: Disjoint circles.
Common mistakes
| Mistake | Why it happens | Fix |
|---|---|---|
| Overlap for "All A are B" only as lens | All A inside B better | A subset of B diagram |
| Double-count overlap in totals | Added n(A)+n(B) | Subtract intersection once |
| Neither forgotten | Only two circles | Rectangle outside = neither |
| Some A are B drawn as all A in B | Over-shading | Only overlap required non-empty |
Quick check
- Shade region: Some A are B.
- 30 like art, 15 like music, 5 both — only art?
- "All A are B" — is every B an A?
- Stretch: 50 children; 35 cricket, 28 football, 20 both — neither sport?
Revision tip: Use two coloured pencil circles — count overlap first in every numbers Venn problem.
Open the Practice tab for graded questions on Venn Diagrams.
Key Takeaways (TL;DR)
- What you'll learn
- Key concepts
- Worked example
- Common mistakes
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