Sets
Subject: Mathematics | Topic: Sets
For TLT, ECT, PST, JEST & All Competitive Exams
The idea of a set is the foundation of modern Mathematics. Functions, relations, probability, algebra and geometry are all built using sets. This topic appears in almost every competitive exam, and the questions are usually about types of sets, subsets, power sets, operations, De Morgan's laws and Venn-diagram problems.
1. What is a Set?
A set is a well-defined collection of distinct objects. "Well-defined" means we can say clearly whether an object belongs to the collection or not. The objects are called elements or members.
- "Vowels of the English alphabet" is a set (a, e, i, o, u).
- "Brave students of a class" is not a set, because "brave" is not well-defined.
Sets are written with capital letters (A, B, C) and elements are written inside curly brackets { }. If x is an element of A we write x ∈ A; if not, x ∉ A. The order of elements does not matter and repeated elements are counted only once: {1, 2, 3} = {3, 1, 2, 2}.
2. Ways of Representing a Set
| Method | Description | Example |
|---|---|---|
| Descriptive | Described in words | A = set of even numbers less than 10 |
| Tabular (Roster) | All elements listed | A = {2, 4, 6, 8} |
| Set-builder | Common property stated | A = {x | x is even, 0 < x < 10} |
3. Standard Number Sets
| Symbol | Name | Elements |
|---|---|---|
| N | Natural numbers | 1, 2, 3, ... |
| W | Whole numbers | 0, 1, 2, 3, ... |
| Z | Integers | ..., -2, -1, 0, 1, 2, ... |
| Q | Rational numbers | p/q, where p, q are integers and q ≠ 0 |
| Q′ | Irrational numbers | Non-terminating, non-repeating decimals (√2, π) |
| R | Real numbers | Q ∪ Q′ |
| P | Prime numbers | 2, 3, 5, 7, 11, ... |
4. Types of Sets
- Empty (Null / Void) set: contains no element. Written as { } or ∅. Example: set of natural numbers between 1 and 2.
- Singleton set: contains exactly one element. Example: {5}.
- Finite set: the elements can be counted. Example: {1, 2, 3}.
- Infinite set: the elements cannot be counted completely. Example: N.
- Equal sets: have exactly the same elements. A = B.
- Equivalent sets: have the same number of elements, i.e. n(A) = n(B), but the elements may differ.
- Universal set (U): the set that contains all elements under discussion; every other set is a subset of it.
- Overlapping sets: have at least one common element but neither is a subset of the other.
- Disjoint sets: have no common element, i.e. A ∩ B = ∅.
5. Subsets
A is a subset of B (A ⊆ B) if every element of A is also in B. A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.
- Every set is a subset of itself: A ⊆ A.
- The empty set is a subset of every set: ∅ ⊆ A.
- If A ⊆ B and B ⊆ A, then A = B.
Number of subsets = 2ⁿ
Number of proper subsets = 2ⁿ − 1
Number of non-empty proper subsets = 2ⁿ − 2
Example: A = {a, b, c}. n = 3, so subsets = 2³ = 8 and proper subsets = 7.
6. Power Set
The set of all subsets of A is called the power set of A, written P(A). It has 2ⁿ elements.
Example: A = {1, 2}. P(A) = { ∅, {1}, {2}, {1, 2} }. So n(P(A)) = 2² = 4.
7. Operations on Sets
Venn Diagram of A and B
| Operation | Symbol | Meaning |
|---|---|---|
| Union | A ∪ B | All elements in A or B or both |
| Intersection | A ∩ B | Elements common to both A and B |
| Difference | A − B | Elements in A but not in B |
| Complement | A′ | Elements of U not in A, i.e. U − A |
Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4}, B = {3, 4, 5, 6}.
- A ∪ B = {1, 2, 3, 4, 5, 6}
- A ∩ B = {3, 4}
- A − B = {1, 2} and B − A = {5, 6}
- A′ = {5, 6, 7, 8}
Note: A − B is not equal to B − A in general. Union and intersection are commutative, but difference is not.
8. Laws of Sets
| Law | Statement |
|---|---|
| Commutative | A ∪ B = B ∪ A; A ∩ B = B ∩ A |
| Associative | (A ∪ B) ∪ C = A ∪ (B ∪ C); (A ∩ B) ∩ C = A ∩ (B ∩ C) |
| Distributive | A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C); A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) |
| Idempotent | A ∪ A = A; A ∩ A = A |
| Identity | A ∪ ∅ = A; A ∩ U = A |
| Complement | A ∪ A′ = U; A ∩ A′ = ∅; (A′)′ = A |
| De Morgan's | (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′ |
9. Counting Formulas (Venn Problems)
n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C)
n(A′) = n(U) − n(A)
If A and B are disjoint: n(A ∪ B) = n(A) + n(B)
Solved Example: In a class of 50 students, 30 like cricket, 25 like football and 10 like both. How many like at least one game? How many like neither?
n(C ∪ F) = 30 + 25 − 10 = 45. Neither = 50 − 45 = 5. Only cricket = 30 − 10 = 20, only football = 25 − 10 = 15.
10. Ordered Pair and Cartesian Product (Introduction)
An ordered pair (a, b) has a fixed order, so (1, 2) ≠ (2, 1). The Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B. If n(A) = m and n(B) = n, then n(A × B) = m × n. In general A × B ≠ B × A.
Example: A = {1, 2}, B = {x, y}. A × B = {(1, x), (1, y), (2, x), (2, y)}, which has 2 × 2 = 4 elements.
11. Quick Exam Tricks
- Number of subsets = 2ⁿ. Number of proper subsets = 2ⁿ − 1.
- ∅ is a subset of every set, but it is an element only if it is written inside the set.
- A ⊆ B means A ∪ B = B and A ∩ B = A.
- Equal sets are always equivalent, but equivalent sets are not always equal.
- Never count a repeated element twice, e.g. {a, a, b} has 2 elements.
Practice MCQs
1. The number of subsets of a set with 4 elements is:
a) 8 b) 16 c) 15 d) 4
2. Which of these is an empty set?
a) {0} b) {∅} c) { } d) {1}
3. If n(A) = 20, n(B) = 15, n(A ∩ B) = 5, then n(A ∪ B) = ?
a) 40 b) 30 c) 35 d) 25
4. (A ∪ B)′ is equal to:
a) A′ ∪ B′ b) A′ ∩ B′ c) A ∩ B d) A − B
5. If n(A) = 3 and n(B) = 4, then n(A × B) = ?
a) 7 b) 12 c) 1 d) 81
Answers: 1-b, 2-c, 3-b, 4-b, 5-b
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