Home › Free notes › Class 11 Mathematics › Sets

Sets – Class 11 Mathematics notes, important questions and MCQs

NCERT Class 11 Mathematics, chapter 1. Notes, key terms, important questions with answers and MCQs.

Chapter overview

A set is a well-defined collection of objects. Sets are combined by union, intersection and difference, and shown with Venn diagrams.

Basics

Roster and set-builder forms; empty set ∅, finite and infinite sets, equal sets.
A ⊆ B if every element of A is in B; a set with n elements has 2ⁿ subsets (power set).
Intervals: (a, b) open, [a, b] closed.

Operations

A ∪ B, A ∩ B, A − B, complement A′ = U − A.
De Morgan's laws: (A ∪ B)′ = A′ ∩ B′; (A ∩ B)′ = A′ ∪ B′.
n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Key terms

Power set: set of all subsets of a set
Disjoint sets: sets with no common element
Universal set: the set containing all elements under consideration, usually denoted U

Formulas

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Number of subsets = 2ⁿ

Important questions with answers

Write the set {x : x is a natural number and x² < 30} in roster form.2 marks

{1, 2, 3, 4, 5}.

If A = {1, 2, 3, 4}, B = {3, 4, 5, 6} and U = {1, 2, …, 8}, find A − B, B − A and (A ∪ B)′.3 marks

A − B = {1, 2}; B − A = {5, 6}; A ∪ B = {1, 2, 3, 4, 5, 6}, so (A ∪ B)′ = {7, 8}.

In a survey of 100 students, 50 read Hindi papers, 40 read English papers and 20 read both. Find how many read (i) at least one, (ii) neither, (iii) only Hindi, (iv) exactly one paper.5 marks

n(H ∪ E) = 50 + 40 − 20 = 70, so (i) 70 read at least one; (ii) 100 − 70 = 30 read neither; (iii) only Hindi = 50 − 20 = 30; (iv) only English = 40 − 20 = 20, so exactly one = 30 + 20 = 50.

Quick revision

∅ is a subset of every set.
De Morgan's laws.
Disjoint ⇒ A ∩ B = ∅.

Exam tips

Proper subsets = 2ⁿ − 1.
For three sets use n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C).

Practice MCQs

Number of subsets of {a, b, c} is
1. 3
2. 6
3. 8
4. 9
Show answer
Answer: (c) 8. 2³.
If A = {1, 2, 3} and B = {3, 4}, A ∩ B is
1. {1, 2}
2. {1, 2, 3, 4}
3. {3}
4. ∅
Show answer
Answer: (c) {3}. Common element.
(A ∪ B)′ equals
1. A′ ∪ B′
2. A − B
3. A ∩ B
4. A′ ∩ B′
Show answer
Answer: (d) A′ ∩ B′. De Morgan.
If n(A) = 20, n(B) = 15, n(A ∩ B) = 5, then n(A ∪ B) is
1. 40
2. 35
3. 30
4. 25
Show answer
Answer: (c) 30. 20 + 15 − 5.
If A = {x : x is a prime less than 10}, then n(A) is
1. 3
2. 4
3. 5
4. 9
Show answer
Answer: (b) 4. {2, 3, 5, 7}.
In a class of 50, 30 like tea, 25 like coffee and 10 like both. How many like neither
1. 0
2. 10
3. 15
4. 5
Show answer
Answer: (d) 5. 50 − 45.
Number of proper subsets of a set with 4 elements is
1. 16
2. 8
3. 15
4. 4
Show answer
Answer: (c) 15. 2⁴ − 1.
If n(U) = 60, n(A) = 25, n(B) = 30 and n(A ∩ B) = 10, then n(A′ ∩ B′) is
1. 5
2. 15
3. 25
4. 35
Show answer
Answer: (b) 15. 60 − (25 + 30 − 10).

All chapters of Class 11 Mathematics

Written by Kundali IQ from standard, well-established facts to help with revision. This is not an official NCERT publication; always read your textbook too.