NCERT Solutions for Exercise 1.5, Class 11, Maths

Exercise 1.5 Sets
Question and Answers
Class 11 – Maths

Class	Class 11
Subject	Mathematics
Chapter Name	Sets
Chapter No.	Chapter 1
Exercise	Exercise 1.5
Category	Class 11 Maths NCERT Solutions

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

(i) A′

(i) A′= U−A

 A’ = {1,2,3,4,5,6,7,8,9} − {1,2,3,4}

A’ ={5,6,7,8,9}

(ii) B′

(ii) B′= U−B 

B’ = {1,2,3,4,5,6,7,8,9} − {2,4,6,8}

B’ = {1, 3, 5, 7, 9}

(iii) (A ∪ C)′

(iii) A U C = {1, 2, 3, 4, 5, 6}

(A ∪ C)′ = U− (A u C)

(A ∪ C)′   = {1,2,3,4,5,6,7,8,9} − {1,2,3,4,5,6}

(A ∪ C)’ = {7, 8, 9}

(iv) (A ∪ B)′

(iv) A U B = {1, 2, 3, 4, 6, 8}

(A ∪ B)′ = U− (A ∪ B)

(A ∪ B)′  ={1,2,3,4,5,6,7,8,9} − {1,2,3,4,5,6,8}

(A ∪ B)’ = {5, 7, 9}

(v) (A′)′

(v) (A’)’ = A = {1, 2, 3, 4}

(vi)  (B – C)′

(vi) B – C = {2, 8}

(B−C)′ = U− (B−C)

(B−C)′ = {1,2,3,4,5,6,7,8,9} − {2,8}

(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 :

(i) A = {a, b, c}

(i) A = {a, b, c}

The complement of set A is the set of all elements of U which are not the elements of A.

A′=U−A

A’ = {a,b,c,d,e,f,g,h} − {a,b,c}

So we get

A’= {d, e, f, g, h}

(ii) B = {d, e, f, g}

(ii)  B = {d, e, f, g}

The complement of set B is the set of all elements of U which are not the elements of B.

B′=U−B

B’ = {a,b,c,d,e,f,g,h} − {d,e,f,g}

So we get

B’ = {a, b, c, h}

(iii) C = {a, c, e, g}

(iii) C = {a, c, e, g}

The complement of set c is the set of all elements of U which are not the elements of C.

C′=U−C

C’ = {a,b,c,d,e,f,g,h} − {a,c,e,g}

So we get

C’ = {b, d, f, h}

(iv) D = { f, g, h, a}

(iv) D = {f, g, h, a}

The complement of set D is the set of all elements of U which are not the elements of D.

D′=U−D

D’ = {a,b,c,d,e,f,g,h} − {f,g,h,a}

So we get

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:

(i) {x : x is an even natural number}

(i) The set of natural number is the universal set.

∴ {x:x is an even natural number}′ = {x:x is an odd natural number}

(ii) { x : x is an odd natural number }

(ii) The set of natural number is the universal set.

 {x: x is an odd natural number}´ =  {x: x is an even natural number}

(iii) {x : x is a positive multiple of 3}

(iii) The set of natural number is the universal set.

{x: x is a positive multiple of 3}´ = {x: x ∈ N and x is not a multiple of 3}

(iv) { x : x is a prime number }

(iv) The set of natural number is the universal set.

{x: x is a prime number}´ ={x: x is a positive composite number and x = 1}

(v) {x : x is a natural number divisible by 3 and 5}

(v) The set of natural number is the universal set.

{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}

(vi) { x : x is a perfect square }

(vi) The set of natural number is the universal set.

{x: x is a perfect square}´ = {x: x ∈ N and x is not a perfect square}

(vii) { x : x is a perfect cube}

(vii) The set of natural number is the universal set.

{x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube}

(viii) { x : x + 5 = 8 }

(viii) The set of natural number is the universal set.

{x: x + 5 = 8}´ = {x: x ∈ N and x ≠ 3}

(ix) { x : 2x + 5 = 9}

(ix) The set of natural number is the universal set.

{x: 2x + 5 = 9}´ = {x: x ∈ N and x ≠ 2}

(x) { x : x ≥ 7 }

(x) The set of natural number is the universal set.

{x: x ≥ 7}´ = {x: x ∈ N and x < 7}

(xi) { x : x ∈ N and 2x + 1 > 10 }

(xii) The set of natural number is the universal set.

{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

(i) (A ∪ B)′ = A′ ∩ B′

(i) A∪B = {2,4,6,8} ∪ {2,3,5,7}

A∪B = {2,3,4,5,6,7,8}

(A∪B)′ = U = A∪B  =  {1, 9}

A′= U−A =  {1,3,5,7,9}

B′= U − B  = {1,4,6,8,9}

A’ ∩ B’ = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}

Therefore, (A U B)’ = A’ ∩ B’.

(ii) (A ∩ B)′ = A′ ∪ B′

(ii) A ∩ B = {2,4,6,8} ∩ {2,3,5,7} = {2}

(A ∩ B)′ = U− A ∩ B = {1,3,4,5,6,7,8,9}

A′= U− A  = {1,3,5,7,9}

B′= U−B  = {1,4,6,8,9}

A′∪ B′ = {1,3,5,7,9} ∪ {1,4,6,8,9} = {1,3,4,5,6,7,8,9}

Therefore, (A ∩ B)’ = A’ U B’.

Question 5 Draw appropriate Venn diagram for each of the following :

(i) (A ∪ B)′,

(i) (A U B)’

(ii) A′ ∩ B′,

(ii) A’ ∩ B’

(iii) (A ∩ B)′,

(iii) (A ∩ B)’

(iv) A′ ∪ B′

(iv) A’ U B’

Question 6 Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?

Answer U is the set of all triangles in the plane

A = Set of triangles different form 60°

A′=U−A = Set of all equilateral triangles

∴ A′ is the set of all equilateral triangles

Question 7 Fill in the blanks to make each of the following a true statement :

(i) A ∪ A′ = . . .

(i) The union of the set and its complement is the universal set

∴ A ∪ A′=U

(ii) φ′ ∩ A = . . .

(ii) ∅′∩ A = U ∩ A = A

∴ ∅′ ∩ A = A

(iii) A ∩ A′ = . . .

(iii) The intersection of the set and its complement is an empty set.

A ∩ A’ = Φ

(iv) U′ ∩ A = . . .

∅ ∩ A = U′ ∩ A = ∅

∴ U’ ∩ A = Φ