Sets
1. What Is a Set?
A SET is a well-defined COLLECTION of distinct objects.
The objects in a set are called ELEMENTS or MEMBERS.
Examples of Well-Defined Sets
- Set of vowels in English: {a, e, i, o, u}
- Set of prime numbers less than 10: {2, 3, 5, 7}
- Set of months with 31 days: {January, March, May, July, August, October, December}
Not a Set (Not Well-Defined)
- 'Set of tall students' — NOT well-defined (tall is subjective).
- 'Set of beautiful paintings' — NOT well-defined (beauty is subjective).
Notation
Sets are usually denoted by CAPITAL letters: A, B, C, ... Elements are written in curly brackets { } separated by commas.
Symbol: ∈ means 'belongs to' or 'is an element of'. ∉ means 'does not belong to'.
Example: If A = {2, 4, 6, 8}, then 4 ∈ A but 5 ∉ A.
2. Representations of Sets
Roster Form (Tabular Form)
List ALL elements separated by commas in curly brackets.
Example: A = {1, 3, 5, 7, 9}.
Set-Builder Form
Describe the COMMON PROPERTY of the elements.
Example: A = {x | x is an odd natural number, x < 10} This reads: 'The set of all x such that x is an odd natural number and x is less than 10.'
Converting Between Forms
Roster to Set-Builder: B = {2, 4, 6, 8, 10} → B = {x | x is an even natural number, x ≤ 10}.
Set-Builder to Roster: C = {x | x is a whole number, x < 5} → C = {0, 1, 2, 3, 4}.
3. Types of Sets
| Type | Definition | Example |
|---|---|---|
| Empty/Null Set | No elements. Denoted by {} or φ (phi) | Set of natural numbers less than 1 |
| Singleton Set | Exactly ONE element | {5}, {0} |
| Finite Set | Countable number of elements | {a, b, c} |
| Infinite Set | Unlimited number of elements | Set of natural numbers N |
| Equal Sets | EXACTLY same elements | {1,2} = {2,1} |
| Equivalent Sets | Same number of elements | {a,b} and {1,2} |
Equal vs Equivalent
- EQUAL sets have the SAME elements. Order does NOT matter.
- {1, 2, 3} = {3, 1, 2} (equal).
- EQUIVALENT sets have the SAME CARDINAL NUMBER.
- A = {p, q, r}, B = {4, 7, 9}. Equivalent? YES (both have 3 elements). Equal? NO.
4. Cardinal Number
The CARDINAL NUMBER of a set is the NUMBER of elements in the set. Denoted by n(A) or |A|.
Examples:
- A = {2, 4, 6, 8} → n(A) = 4
- B = {} → n(B) = 0
- C = {x | x is a letter in 'MATHEMATICS'} → n(C) = 8 (M, A, T, H, E, I, C, S — repeated letters count once)
5. Subsets and Supersets
A is a SUBSET of B if EVERY element of A is also in B. Written as A ⊆ B. If A is not a subset of B: A ⊈ B.
Properties
- Every set is a subset of itself: A ⊆ A.
- Empty set is a subset of EVERY set: φ ⊆ A for any set A.
- If A ⊆ B and B ⊆ A, then A = B.
Proper Subset (⊂): A ⊂ B means A ⊆ B but A ≠ B.
Superset: If A ⊆ B, then B is a SUPERSET of A. Written as B ⊇ A.
Power Set: The set of ALL subsets of a given set. If A = {1, 2}, P(A) = {φ, {1}, {2}, {1, 2}}. If n(A) = m, then n(P(A)) = 2ᵐ.
6. Venn Diagrams
Venn diagrams use RECTANGLES and CIRCLES to represent sets visually.
- Universal set (U) = RECTANGLE
- Sets = CIRCLES inside the rectangle
Operations on Sets
| Operation | Symbol | Meaning | Venn Diagram |
|---|---|---|---|
| Union | A ∪ B | Elements in A OR B (or both) | Shade both circles |
| Intersection | A ∩ B | Elements in BOTH A and B | Shade overlap region |
| Complement | A' or Aᶜ | Elements NOT in A (in U) | Shade everything outside A |
| Difference | A - B | Elements in A but NOT in B | Shade A excluding overlap |
Venn Diagram Rules
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
- n(A - B) = n(A) - n(A ∩ B)
- n(A') = n(U) - n(A)
7. Worked Examples (ICSE Focus)
Example 1 (ICSE 2024, 3 marks)
If U = {1, 2, 3, ..., 10}, A = {2, 4, 6, 8, 10}, B = {5, 6, 7, 8, 9, 10}. Find: (i) A ∪ B (ii) A ∩ B (iii) A' (iv) n(A - B).
Solution: (i) A ∪ B = {2, 4, 5, 6, 7, 8, 9, 10} (ii) A ∩ B = {6, 8, 10} (iii) A' = {1, 3, 5, 7, 9} (iv) A - B = {2, 4}. n(A - B) = 2.
Example 2 (ICSE 2023, 2 marks)
In a class of 40 students, 25 like cricket and 20 like football. 10 like both. How many like NEITHER?
Solution: n(U) = 40, n(C) = 25, n(F) = 20, n(C ∩ F) = 10. n(C ∪ F) = 25 + 20 - 10 = 35. Students who like neither = n(U) - n(C ∪ F) = 40 - 35 = 5.
8. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Set notation and types | 2 marks | High |
| Subsets and cardinal number | 2 marks | Medium |
| Venn diagram operations | 3-4 marks | Very High |
| Word problems using sets | 3-4 marks | High |
Common Mistakes
- Writing repeated elements in roster form (each element appears ONCE only).
- Confusing ∈ and ⊆. ∈ is for ELEMENTS, ⊆ is for SETS.
- Forgetting to subtract intersection in union formula.
- Writing φ as {φ} — WRONG. φ is empty set, {φ} is a set containing empty set.
Self-Test (5 Questions)
Q1. Which is a well-defined set? (1 mark)
- A) Set of good books
- B) Set of prime numbers
- C) Set of tall boys
- D) Set of beautiful flowers
Q2. Write in set-builder form: A = {4, 6, 8, 9, 10}. (2 marks)
Q3. If n(A) = 5, n(B) = 7, n(A ∩ B) = 3, find n(A ∪ B). (2 marks)
Q4. If U = {x | x ∈ N, x ≤ 12} and A = {x | x is a prime number ≤ 12}, find A'. (2 marks)
Q5. In a group of 60 people, 35 speak Hindi, 30 speak English, and 15 speak both. How many speak NEITHER language? (3 marks)
Answers
A1. B) Set of prime numbers. A2. A = {x | x is a composite number, 4 ≤ x ≤ 10}. (Or similar valid description.) A3. 9. (n(A ∪ B) = 5 + 7 - 3 = 9.) A4. A' = {1, 4, 6, 8, 9, 10, 12}. (U = {1,...,12}, primes ≤ 12 = {2,3,5,7,11}.) A5. 10. (n(U)=60, n(H)=35, n(E)=30, n(H∩E)=15. n(H∪E)=35+30-15=50. Neither = 60-50=10.)
