1. Basics
- A set is a well-defined collection. Roster form {1, 2, 3}; set-builder form {x : x is a natural number less than 4}.
- Empty set ∅ has no elements; a singleton has one.
- n(A) = number of elements (cardinality).
2. Subsets and power set
| Quantity | Formula (n elements) |
|---|---|
| Subsets / elements of the power set | 2ⁿ |
| Proper subsets | 2ⁿ − 1 |
| Non-empty proper subsets | 2ⁿ − 2 |
(ASO 2021) 5 elements → power set has 2⁵ = 32 elements.
3. Operations
- A ∪ B: in A or B. A ∩ B: in both. A − B: in A but not B. A′: not in A.
- De Morgan: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′.
4. Counting formulas
- n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
- Only A = n(A) − n(A ∩ B)
- Neither = Total − n(A ∪ B)
- n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A∩B) − n(B∩C) − n(C∩A) + n(A∩B∩C)
5. Solved examples
n(A) = 40, n(B) = 30, n(A ∩ B) = 10 → n(A ∪ B) = 60.
(ASO 2021) 500 students, 475 Hindi, 200 English (all speak one) → Hindi only = 500 − 200 = 300.
100 people: 60 tea, 50 coffee, 20 both → neither = 100 − 90 = 10.
Of 50, 30 like cricket, 25 football, everyone likes at least one → both = 55 − 50 = 5.
6. Venn diagram method
Always fill the innermost region (all three, or both) first, then work outwards, subtracting.
Solved examples by question type
Each type that comes in the ASO paper, worked step by step. Try it yourself first, then read the solution.
Basic level
Example 1. If a set has 6 elements, its power set has how many elements?
(a) 36 (b) 64 (c) 63 (d) 12
Answer: (b) 64
Solution: Power set has 2^n = 2^6 = 64 elements.
Example 2. If n(A) = 49, n(B) = 22 and n(A ∩ B) = 9, then n(A ∪ B) = ?
(a) 62 (b) 71 (c) 53 (d) 80
Answer: (a) 62
Solution: n(A ∪ B) = 49 + 22 − 9 = 62.
Example 3. If n(A) = 65, n(B) = 62 and n(A ∩ B) = 38, then n(A ∪ B) = ?
(a) 89 (b) 51 (c) 127 (d) 165
Answer: (a) 89
Solution: n(A ∪ B) = 65 + 62 − 38 = 89.
Example 4. A set has 6 elements. The number of its proper subsets is:
(a) 64 (b) 36 (c) 12 (d) 63
Answer: (d) 63
Solution: Subsets = 2^6 = 64; proper subsets exclude the set itself → 63.
Example 5. A set has 7 elements. The number of its subsets is:
(a) 49 (b) 128 (c) 14 (d) 127
Answer: (b) 128
Solution: Subsets = 2^7 = 128.
ASO exam level
Example 6. In a group of 500 people, 475 speak Hindi and 200 speak English. Everyone speaks at least one of the two. How many speak both?
(a) 300 (b) 175 (c) 185 (d) 25
Answer: (b) 175
Solution: n(H ∪ E) = n(H) + n(E) − n(both) → 500 = 475 + 200 − x → x = 175.
Example 7. Out of 80 students, 45 like tea, 40 like coffee and 15 like neither. How many like only tea?
(a) 20 (b) 26 (c) 45 (d) 25
Answer: (d) 25
Solution: Like at least one = 65. Both = 45 + 40 − 65 = 20. Only tea = 45 − 20 = 25.
Example 8. In how many ways can a team of 11 be chosen from 15 players?
(a) 2,730 (b) 1,365 (c) 165 (d) 3,003
Answer: (b) 1,365
Solution: 15C11 = 15C4 = 1365.
Example 9. In how many ways can the letters of the word 'ROOT' be arranged?
(a) 13 (b) 12 (c) 24 (d) 6
Answer: (b) 12
Solution: 4! divided by factorials of repeated letters (O×2) = 12.
Example 10. In a group of 300 people, 193 like tea and 272 like coffee. Everyone likes at least one. How many like both?
(a) 165 (b) 175 (c) 28 (d) 107
Answer: (a) 165
Solution: Both = 193 + 272 − 300 = 165.
Example 11. In a class, 53 students play cricket, 65 play football and 39 play both. How many play cricket only?
(a) 79 (b) 14 (c) 26 (d) 53
Answer: (b) 14
Solution: Cricket only = 53 − 39 = 14.
Advanced level
Example 12. At a meeting, each of 20 people shakes hands with every other person once. The total number of handshakes is:
(a) 190 (b) 380 (c) 210 (d) 400
Answer: (a) 190
Solution: Handshakes = nC2 = 20×19/2 = 190.
Example 13. Out of 170 students, 42 study Hindi, 62 study Odia and 37 study both. How many study neither?
(a) 67 (b) 103 (c) 66 (d) 140
Answer: (b) 103
Solution: At least one = 42 + 62 − 37 = 67. Neither = 170 − 67 = 103.
Topic-wise practice for Sets (Quant): Operations and Venn diagrams. Useful for OPSC ASO, OSSC, OSSSC and other Odisha exams.
Question types covered
- Types of sets; roster and set-builder form
- Union, intersection, difference, complement
- Number of subsets and power set
- n(A ∪ B) formula
- Venn diagram word problems