How many subsets does the set A = {1, 2, 3} have?

8. A set with n elements has 2ⁿ subsets: 2³ = 8.

The number of proper subsets of a set with 4 elements is:

15. Proper subsets = 2ⁿ − 1 = 16 − 1 = 15.

If n(A) = 20, n(B) = 15 and n(A ∩ B) = 5, then n(A ∪ B) is:

30. n(A ∪ B) = 20 + 15 − 5 = 30.

The set {x | x ∈ R, 2 < x ≤ 5} in interval form is:

(2, 5]. 2 is excluded (round bracket) and 5 is included (square bracket).

A ∩ B′. A − B contains elements of A that are not in B, i.e. A ∩ B′.

Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 1: Sets and Relations

Short notes, key concepts, quick revision and practice MCQs with answers

A set is a well-defined collection of objects. The chapter covers ways of writing sets, intervals on the real line, types of sets, operations such as union, intersection and complement, counting the elements of combined sets, ordered pairs, Cartesian products and relations.

Key concepts: Sets and Relations

Set and its representation: A collection of well-defined objects. Roster (list) form: A = {1, 2, 3}. Set-builder form: A = {x | x ∈ N, x ≤ 3}. The set theory was developed by Georg Cantor.
Intervals: Open (a, b) = {x | a < x < b}; closed [a, b] = {x | a ≤ x ≤ b}; semi-open [a, b) and (a, b]. A round bracket leaves the end point out, a square bracket includes it.
Types of sets: Empty (null) set ∅ has no element; singleton has one element; finite and infinite sets; equal sets have the same elements; equivalent sets have the same number of elements.
Subsets and power set: A ⊂ B if every element of A is in B. A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets. The power set P(A) is the set of all subsets of A.
Operations on sets: Union A ∪ B, intersection A ∩ B, difference A − B = {x | x ∈ A, x ∉ B}, complement A′ = U − A. De Morgan's laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.
Number of elements: n(A ∪ B) = n(A) + n(B) − n(A ∩ B). For three sets: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(A ∩ C) + n(A ∩ B ∩ C).
Ordered pair and Cartesian product: (a, b) = (x, y) only if a = x and b = y. A × B = {(a, b) | a ∈ A, b ∈ B}. If n(A) = p and n(B) = q then n(A × B) = pq.
Relations: A relation R from A to B is a subset of A × B. Domain = set of first components, range = set of second components. On a set A, R is reflexive, symmetric or transitive; if all three, it is an equivalence relation.

Quick revision: important points

Number of subsets of an n-element set = 2ⁿ; proper subsets = 2ⁿ − 1.
If A and B are disjoint, n(A ∪ B) = n(A) + n(B).
A − B = A ∩ B′; A ∪ A′ = U and A ∩ A′ = ∅.
n(A × B) = n(A) × n(B), but A × B ≠ B × A in general.
Number of relations from A to B = 2^(pq) where n(A) = p, n(B) = q.
Reflexive: (a, a) ∈ R for all a. Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R. Transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R.

Practice MCQs with answers

How many subsets does the set A = {1, 2, 3} have?
1. 3
2. 6
3. 8
4. 9
Show answer
Answer: C. 8
A set with n elements has 2ⁿ subsets: 2³ = 8.
The number of proper subsets of a set with 4 elements is:
1. 16
2. 15
3. 8
4. 4
Show answer
Answer: B. 15
Proper subsets = 2ⁿ − 1 = 16 − 1 = 15.
If n(A) = 20, n(B) = 15 and n(A ∩ B) = 5, then n(A ∪ B) is:
1. 40
2. 35
3. 30
4. 25
Show answer
Answer: C. 30
n(A ∪ B) = 20 + 15 − 5 = 30.
The set {x | x ∈ R, 2 < x ≤ 5} in interval form is:
1. (2, 5)
2. [2, 5]
3. (2, 5]
4. [2, 5)
Show answer
Answer: C. (2, 5]
2 is excluded (round bracket) and 5 is included (square bracket).
A − B is equal to:
1. A ∩ B
2. A ∩ B′
3. A′ ∩ B
4. A ∪ B′
Show answer
Answer: B. A ∩ B′
A − B contains elements of A that are not in B, i.e. A ∩ B′.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters