Sets
1. Definition of a Set
A set is a well-defined collection of distinct objects.
Elements are listed inside curly braces { }.
Examples: A = {1, 2, 3, 4, 5} B = {x : x is a vowel in English} = {a, e, i, o, u} C = {x : x is an integer and —3 ≤ x < 4} = {—3, —2, —1, 0, 1, 2, 3}
'Well-defined means we can CLEARLY determine whether any given object belongs to the set or not.'
2. Notation and Terminology
| Symbol | Meaning |
|---|---|
| ∈ | 'belongs to' or 'is an element of' |
| ∉ | 'does not belong to' |
| n(A) | Cardinal number (number of elements in set A) |
| φ or {} | Empty set (null set) — no elements |
| U or ξ | Universal set — contains ALL elements under consideration |
Example: If A = {1, 2, 3, 4, 5}, then 3 ∈ A, 8 ∉ A, n(A) = 5.
3. Types of Sets
| Type | Definition | Example |
|---|---|---|
| Empty set | No elements | Set of months with 32 days = φ |
| Singleton | Exactly ONE element | {5} |
| Finite set | Countable number of elements | {a, b, c} — 3 elements |
| Infinite set | Endless number of elements | Set of natural numbers N = {1, 2, 3, ...} |
| Equal sets | EXACTLY the same elements | A = {1, 2}, B = {2, 1} — A = B |
| Equivalent sets | SAME number of elements | A = {a, b, c}, B = {4, 7, 9} — n(A) = n(B) = 3 |
'Equal sets are ALWAYS equivalent, but equivalent sets are NOT always equal.'
4. Subsets and Supersets
Subset: A ⊆ B means EVERY element of A is also in B. Proper subset: A ⊂ B means A ⊆ B but A ≠ B. Superset: B ⊇ A means B contains A.
If A = {2, 4} and B = {1, 2, 3, 4, 5}: A ⊂ B (proper subset). B ⊃ A (proper superset).
'Every set is a subset of ITSELF. The empty set is a subset of EVERY set.'
5. Cardinal Number
n(A) = number of elements in set A.
If A = {x : x is a prime number less than 10} = {2, 3, 5, 7}, then n(A) = 4.
Formula: If A and B are disjoint sets, n(A ∪ B) = n(A) + n(B).
6. Set Operations — Venn Diagrams
Union (A ∪ B)
Set of elements that belong to A OR B (or both). A ∪ B = {x : x ∈ A or x ∈ B} n(A ∪ B) = n(A) + n(B) — n(A ∩ B)
Intersection (A ∩ B)
Set of elements that belong to BOTH A and B. A ∩ B = {x : x ∈ A and x ∈ B}
Complement (A')
Set of elements in the universal set that are NOT in A. A' = {x : x ∈ U and x ∉ A} n(A) + n(A') = n(U)
Difference (A — B)
Set of elements that belong to A but NOT to B.
7. Disjoint Sets
Sets A and B are DISJOINT if they have NO elements in common. A ∩ B = φ and n(A ∩ B) = 0.
8. Properties of Set Operations
| Property | Union | Intersection |
|---|---|---|
| Commutative | A ∪ B = B ∪ A | A ∩ B = B ∩ A |
| Associative | (A∪B)∪C = A∪(B∪C) | (A∩B)∩C = A∩(B∩C) |
| Identity | A ∪ φ = A | A ∩ U = A |
| Idempotent | A ∪ A = A | A ∩ A = A |
| Distributive | A∪(B∩C) = (A∪B)∩(A∪C) | A∩(B∪C) = (A∩B)∪(A∩C) |
De Morgan's Laws: (A ∪ B)' = A' ∩ B' (A ∩ B)' = A' ∪ B'
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Order matters in a set' | {1, 2, 3} = {3, 1, 2}. Order DOES NOT matter |
| 'Repeating elements increase the count' | Sets DO NOT have repeated elements. {1, 1, 2} = {1, 2} |
| 'φ and {0} are the same' | φ has NO elements. {0} has ONE element (0). They are DIFFERENT |
| 'Confusing ⊆ and ⊂' | A ⊆ B means A is a subset (could be equal). A ⊂ B means PROPER subset (not equal) |
ICSE Exam Focus (5–6 marks)
- 2-mark questions: Types of sets, identifying subsets
- 3-mark questions: Set operations with Venn diagrams
- 4-mark questions: Cardinal number word problems (survey data)
- 6-mark questions: De Morgan's laws verification with Venn diagrams
Self-Test
Q1. If A = {x : x is a letter in the word 'MATHEMATICS'}, list the elements of A and find n(A). A1. A = {M, A, T, H, E, I, C, S}. Note: repeated letters counted once. n(A) = 8.
Q2. Are {1, 2, 3} and {3, 2, 1} equal sets? Are they equivalent? A2. They have exactly the same elements (order irrelevant). Yes, they are EQUAL. They are also equivalent (n = 3).
Q3. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {2, 3, 5, 7}. Find A ∪ B, A ∩ B, A'. A3. A ∪ B = {1, 2, 3, 5, 7, 9}. A ∩ B = {3, 5, 7}. A' = {2, 4, 6, 8, 10}.
Q4. In a class of 50 students, 35 like Maths and 30 like Science. Each student likes at least one subject. How many like BOTH? A4. n(M∪S) = 50, n(M) = 35, n(S) = 30. n(M∩S) = n(M) + n(S) — n(M∪S) = 35 + 30 — 50 = 15. Answer: 15 students.
Q5. Verify De Morgan's law: (A ∪ B)' = A' ∩ B' for A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, U = {1, 2, 3, 4, 5, 6, 7, 8}. A5. A∪B = {1,2,3,4,5,6}. (A∪B)' = {7,8}. A' = {5,6,7,8}, B' = {1,2,7,8}. A'∩B' = {7,8}. Hence verified.
Q6. Write all subsets of {a, b, c}. A6. φ, {a}, {b}, {c}, {a,b}, {a,c}, {b,c}, {a,b,c}. Total = 2³ = 8 subsets.
