Venn Diagrams — SSC CGL Reasoning

Venn questions come in two flavours. Relationship diagrams ask which picture models three classes — is one inside another, do they overlap, or are they separate? Numerical Venns give you numbers in the regions and ask how many fall in an overlap or in none. The first is answered by asking "can a single thing belong to both?"; the second by inclusion–exclusion.

1. What SSC actually asks

Tier 1: 1–2 Q · Tier 2: 1–2 Q. Types: choose the correct relationship diagram for three (sometimes four) given terms, and the numerical two- or three-set problem — "how many play only cricket / both / neither" from given totals.

2. The three basic relationships

The deciding question is always: can one item be both at once?

• Table, Chair, Furniture → Furniture is the big circle; Table and Chair are two separate circles inside (a table is never a chair).
• Women, Mothers, Doctors → Mothers inside Women (every mother is a woman); Doctors overlapping (a doctor may or may not be a woman).
• Earth, Sun, Moon → three separate circles (distinct, non-overlapping).
• India, Asia, World → nested: India ⊂ Asia ⊂ World.

3. Numerical Venn — two sets

For sets A and B:

$|A\cup B|=|A|+|B|-|A\cap B|$

• Only A $=|A|-|A\cap B|$; only B $=|B|-|A\cap B|$.
• Neither $=\text{Total}-|A\cup B|$.

100 students: 60 cricket, 40 football, 20 both. At least one $=60+40-20=80$; neither $=20$; only cricket $=40$.

4. Numerical Venn — three sets

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|A\cap C|+|A\cap B\cap C|$

Work inside-out: fill the triple overlap first, then subtract it from each pair region, then subtract those from each single set — the "only" regions are what remain.

Survey of 60: A 25, B 26, C 26; A∩B 9, B∩C 11, A∩C 8, all three 3. At least one $=77-28+3=52$; read none $=60-52=8$.

5. Reading a pre-filled diagram

When numbers are already placed in each region, don't apply formulas — just add the regions the question asks for:

• "Only maths" = the maths-only region alone.
• "Maths and science but not history" = the single region in that specific overlap.
• Always match the wording ("only", "exactly two", "at least one") to the exact regions.

6. Solved PYQ-style examples

Q1. Which diagram best represents Table, Chair, Furniture? Solution. Furniture as the enclosing circle; Table and Chair as two separate circles inside it.

Q2. In a class of 100, 60 play cricket, 40 play football and 20 play both. How many play neither? Solution. At least one $=60+40-20=80$ → neither $=100-80=$ 20.

Q3. 40 know English, 30 know Hindi, 15 know both. How many know only English? Solution. $40-15=$ 25.

Q4. In a survey of 60 people: A 25, B 26, C 26; A∩B 9, B∩C 11, A∩C 8, all three 3. How many read none? Solution. At least one $=77-28+3=52$; none $=60-52=$ 8.

Q5. Which relationship fits India, Asia, World? Solution. Nested circles: India inside Asia inside World.

7. Exam protocol

1. Relationship diagrams: ask "can one item be both?" — yes → overlap, always → subset, never → disjoint.
2. Two sets: $|A\cup B|=|A|+|B|-|A\cap B|$; "neither" = total − union.
3. Three sets: fill the centre first, work outward; use the full inclusion–exclusion.
4. Pre-filled diagram: add only the regions named — read "only/exactly/at least" precisely.
5. Sanity-check with reality: a doctor can be a woman (overlap), but a table can't be a chair (disjoint inside furniture).