Venn Diagrams

A Venn diagram uses overlapping circles to show relationships between sets.

Two-set Venn (sets A and B):

• A ∪ B (union): everything in A or B (or both).
• A ∩ B (intersection): only the overlap.
• A − B: in A but not in B.
• |A ∪ B| = |A| + |B| − |A ∩ B|.

Three-set Venn (sets A, B, C):

• |A ∪ B ∪ C| = |A| + |B| + |C| − |A∩B| − |B∩C| − |A∩C| + |A∩B∩C|.
  This is the inclusion-exclusion principle.

Regions of a 3-set Venn — 7 regions for the union, plus 1 for "outside all":

1. Only A
2. Only B
3. Only C
4. A∩B only (not C)
5. B∩C only (not A)
6. A∩C only (not B)
7. A∩B∩C (all three)
8. None (outside all circles)

RRB-style question typically gives:

• Total surveyed people
• Number liking each of 2 or 3 items
• Pairwise overlaps
• All-three overlap
• Asks: how many like exactly one? exactly two? none?

Strategy:

1. Draw the diagram and start from the centre (all-three overlap) and work outward.
2. Subtract larger overlaps from smaller ones to find "exclusive" regions.
3. "Only A" = |A| − |A∩B| − |A∩C| + |A∩B∩C|.
4. "Exactly two" = sum of pairwise overlaps minus 3 × triple overlap.

Relational Venn (logic questions): given categories like "doctors", "men", "graduates", and asked to pick which Venn diagram represents them. Use:

• If A is a subset of B → A inside B.
• If A and B overlap partially → overlapping circles.
• If A and B are disjoint → separate circles.

Example 1 — Two-set:
In a class of 50, 30 like cricket and 25 like football. 10 like both. How many like neither?
Method: |C ∪ F| = 30 + 25 − 10 = 45. Neither = 50 − 45 = 5.

Example 2 — Three-set:
In a survey of 100 people: 60 like tea, 50 like coffee, 40 like milk. 30 like tea+coffee, 20 like coffee+milk, 25 like tea+milk. 15 like all three. How many like only tea?
Method: Only tea = Tea − (T∩C) − (T∩M) + (T∩C∩M) = 60 − 30 − 25 + 15 = 20.

Example 3 — Exactly two:
From the same data, how many like exactly two?
Method: Exactly two = (T∩C) + (C∩M) + (T∩M) − 3×(T∩C∩M) = 30 + 20 + 25 − 3×15 = 75 − 45 = 30.

Example 4 — None:
From above, how many liked none?
Method: Total who like at least one = 60+50+40 − 30−20−25 + 15 = 150 − 75 + 15 = 90.
None = 100 − 90 = 10.

Example 5 — Relational:
Which Venn diagram represents "Dogs, Mammals, Animals"?
Method: Dogs ⊂ Mammals ⊂ Animals. Three concentric circles: Dogs inside Mammals inside Animals.

Example 6 — Disjoint:
"Tigers, Lions, Sparrows".
Method: Tigers and Lions are disjoint (different species) but both are subsets of a wider "Animals" category. Sparrows are also separate. Three separate circles (perhaps inside a larger one labelled Animals).

Trap: in the three-set formula, the +15 at the end (for triple-overlap) is added, not subtracted — it was over-subtracted three times by the pairwise terms.