Chapter 1 of the Class 11 NCERT Mathematics textbook, "Sets," introduces the basic concepts of sets, including definitions, types of sets, and set operations. Exercise 1.5 focuses on solving problems related to the operations and properties of sets.
NCERT Solutions for Class 11 - Mathematics - Chapter 1 Sets - Exercise 1.5
This section provides detailed solutions for Exercise 1.5 from Chapter 1 of the Class 11 NCERT Mathematics textbook. The exercise involves problems related to the operations on sets, such as union, intersection, and complement, as well as using Venn diagrams to solve problems.
Question 1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find:
(a) A'
Solution:
We know that this is the complement of set A i.e, it is the subset of U.
So, A' = {5, 6, 7, 8, 9}
(b) B'
Solution:
We know that this is the complement of set B i.e, it is the subset of U.
So, B' = {1, 3, 5, 7, 9}
(c) (A ∪ C)'
Solution:
This is the complement of union of set A and set C i.e, U - (A∪ C)
So, A∪ C = {1, 2, 3, 4, 5, 6}
=> U - (A ∪ C)
So, (A∪ C)' = {7, 8, 9}
(d) (A ∪ B)'
Solution:
This is complement of union of set A and set B i.e, U- (A∪B)
So, A∪ B = {1, 2, 3, 4, 6, 8}
=> U - (A ∪ B)
So, (A ∪ B)' = {5, 7, 9}
(e) (A')'
Solution:
This is the complement of set A i.e, (A')' = A
So, (A')' = {1, 2, 3, 4}
(f) (B - C)'
Solution:
(B - C) = elements in B but not in C
(B - C) = {2, 8}
=> U - (B - C)
So, (B - C)' = {1, 3, 4, 5, 6, 7, 9}
Question 2. If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets :
(a) A = {a, b, c}
Solution:
Complement of set A = A'
A' = U - A
A' = {a, b, c, d, e, f, g, h} - {a, b, c}
A' = {d, e, f, g, h}
(b) B = {d, e, f, g}
Solution:
Complement of set B = B'
B' = U - B
B' = {a, b, c, d, e, f, g, h} - {d, e, f, g}
B' = {a, b, c, h}
(c) C = {a, c, e, g}
Solution:
Complement of set C = C'
C' = U - C
C' = {a, b, c, d, e, f, g, h} - {a, c, e, g}
C' = {b, d, f, h}
(d) D = {f, g, h, a}
Solution:
Complement of set D = D'
D' = U - D
D' = {a, b, c, d, e, f, g, h} - {f, g, h, a}
D' = {b, c, d, e}
Question 3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
U = N: set of natural numbers
(a) {x : x is an even natural number}
=> {x : x is an odd natural number}
(b) {x : x is an odd natural number}
=>{x : x is an even natural number}
(c) {x : x is a positive multiple of 3}
=>{x : x∈N and x is not a multiple of 3}
(d) {x : x is a prime number}
=> {x : x is a positive composite number and x=1}
(e) {x : x is a natural number divisible by 3 and 5}
=> {x : x is a natural number that is not divisible by 3 or 5}
(f) {x : x is a perfect square}
=> {x : x∈N and x is not perfect square}
(g) {x : x is a perfect cube}
=> {x : x∈N and x is not perfect cube}
(h) {x : x + 5 = 8}
=> {x : x∈N and x≠3}
(i) {x : 2x + 5 = 9}
=> {x : x∈N and x≠2}
(j) {x : x ≥ 7}
=> {x : x∈N and x<7}
(k) {x : x ∈ N and 2x + 1 > 10}
=> {x : x∈N and x≤ 9/2}
Question 4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = { 2, 3, 5, 7}. Verify that
(a) (A ∪ B)'= A' ∩ B'
Solution:
=> (A ∪ B)'= U - (A∪B)
=> {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2, 3, 4, 5, 6, 7, 8}
=> (A∪B)' = {1, 9}
A' ∩ B' = (U - A) ∩ (U - B)
=> {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9}
=> A' ∩ B' = {1, 9}
Hence, Verified!!! (A∪ B)' = A' ∩ B'
(b) (A ∩ B)′ = A′ ∪ B′
=> (A ∩ B)' = U - (A ∩ B)
=> {1, 2, 3, 4, 5, 6, 7, 8, 9} - {2}
=> (A ∩ B)′ = {1, 3, 4, 5, 6, 7, 8, 9}
A' ∪ B'= (U - A) ∪ (U - B)
=> {1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9}
=> A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}
Hence, Verified!!! (A ∩ B)′ = A′ ∪ B′
Question 5. Draw appropriate Venn diagram for each of the following:
(a) (A ∪ B)′=
(b) A' ∩ B'
(c) (A ∩ B)′
(d) A′ ∪ B′
Question 6. Let U be the set of all triangles in a plane. If A is the set of all triangles with atleast one angle different from 60°, what is A′?
Solution:
U = set of all triangles in plane
A = set of all triangles with at least one angle different from 60°
A' = set of all triangles with no angle different from 60° i.e, set of all triangles with all angles 60°
A' is the set of all equilateral triangle.
Question 7. Fill in the blanks to make each of the following a true statement :
Solution:
(a) A ∪ A′ = U
(b) ∅′ ∩ A = A
(c) A ∩ A′ = ∅
(d) U′ ∩ A =.∅
Summary
Chapter 1 of the NCERT Class 11 Mathematics textbook covers the concept of Sets. In Exercise 1.5, students learn to solve problems involving the operations of union, intersection, and complement of sets, as well as the properties of these operations. The exercises also introduce students to the concept of the universal set and the power set of a given set. Overall, this exercise helps students develop a deeper understanding of set theory and its applications.