11  Set Theory

11.1 Introduction

Set Theory is the mathematical language of collections.
It provides a foundation for counting, probability, logic, and data analysis.
In competitive exams, set theory appears in the form of Venn diagrams, union–intersection problems, and word problems involving “at least one,” “only,” and “none.”

11.2 1) Basics of Sets

  • Set: Well-defined collection of distinct objects.
  • Element (∈): Member of a set.
  • Empty set (∅): No elements.
  • Universal set (U): Set containing all elements under discussion.
  • Subset (⊆): A ⊆ B if every element of A is also in B.
  • Proper subset (⊂): A ⊆ B but A ≠ B.
  • Power set: Set of all subsets. If |A|=n, then |P(A)|=2^n.
  • Cardinality |A|: Number of elements in set A.

11.3 2) Representation

  • Roster form: A={1,2,3,4}
  • Rule form: A={x: x is a natural number ≤4}
  • Venn diagrams: Graphical circles representing sets.

11.4 3) Operations on Sets

  • Union: A ∪ B = {x: x∈A or x∈B}
  • Intersection: A ∩ B = {x: x∈A and x∈B}
  • Difference: A − B = {x: x∈A but not in B}
  • Complement: A′ = U − A

11.5 4) Laws of Sets

  • Idempotent: A∪A=A, A∩A=A
  • Commutative: A∪B=B∪A; A∩B=B∩A
  • Associative: (A∪B)∪C=A∪(B∪C)
  • Distributive: A∩(B∪C)=(A∩B)∪(A∩C)
  • De Morgan’s laws:
    (A∪B)′=A′∩B′,
    (A∩B)′=A′∪B′

11.6 5) Counting with Sets

11.6.1 5.1 Two sets

\[ |A \cup B| = |A| + |B| - |A \cap B| \]

11.6.2 5.2 Three sets

\[ |A \cup B \cup C| = |A|+|B|+|C| - |A\cap B| - |B\cap C| - |C\cap A| + |A\cap B\cap C| \]

11.7 6) Solved Examples

Ex 1: In a class, 20 like maths, 15 like science, 10 like both. How many like at least one?
=20+15−10=25.

Ex 2: Out of 200 people, 80 like coffee, 120 like tea, 60 like both. How many like neither?
Coffee ∪ Tea = 80+120−60=140.
Neither = 200−140=60.

Ex 3: In survey, 60% like A, 50% like B, 30% like both. Probability random person likes at least one?
=0.6+0.5−0.3=0.8=80%.

Ex 4: From 1–20, let A={multiples of 2}, B={multiples of 3}. Find |A∪B|.
|A|=10, |B|=6, |A∩B|=3 ⇒ |A∪B|=13.

11.8 7) Word Problem Patterns

  • At least one = total − none.
  • Exactly one = sum of singles − 2×intersection.
  • Only A = |A|−|A∩B|.
  • Neither = U − (A∪B).

11.9 8) Venn Diagram Applications

  • 2-set problems: Typically about students liking two subjects.
  • 3-set problems: Use inclusion–exclusion step by step.
  • Complement: Shade outside the circle to represent “not.”

11.10 9) Practice Set – Level 1

  1. In 60 students, 25 play cricket, 20 football, 10 both. How many play neither?
  2. 50 students: 30 passed maths, 20 passed English, 10 both. How many failed both?
  3. Out of 100, 40 read newspaper A, 50 read B, 20 both. How many read exactly one?
  4. In set A={1,2,3,4}, B={3,4,5,6}, find A∪B and A∩B.
  5. If U={1,2,…,10}, A={2,4,6,8,10}, find A′.

11.11 Practice Set – Level 2

  1. In 120 people, 60 like tea, 45 like coffee, 20 like both. Find neither.
  2. 100 students: 60 passed Maths, 50 Physics, 30 both. How many passed at least one?
  3. In a survey, 200 people: 80 use Brand A, 70 Brand B, 60 Brand C, 20 all three, 30 A∩B, 25 B∩C, 15 C∩A. Find number using none.
  4. 500 voters: 300 like candidate A, 250 like B, 100 both. How many like exactly one?
  5. If A={x: x<10, x∈N}, B={x: x is even}, find A−B.

11.12 10) Common Mistakes

  • Forgetting to subtract intersection when counting union.
  • Mixing up “exactly one” and “at least one.”
  • Overcounting in 3-set Venn diagrams.
  • Misusing complement (forgetting universal set definition).

11.13 Answer Key (outline)

Level 1: (1) 25, (2) 10, (3) 50, (4) A∪B={1,2,3,4,5,6}, A∩B={3,4}, (5) {1,3,5,7,9}.
Level 2: (6) 35, (7) 80, (8) 200−(80+70+60−30−25−15+20)=200−160=40, (9) 350, (10) {1,3,5,7,9}.

11.14 Summary

  • Sets = structured collections.
  • Use inclusion–exclusion for counts.
  • Represent with Venn diagrams.
  • Translate word problems into set equations.
  • Powerful tool for probability, DI, and logical reasoning.