**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*: 2*x* + 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 2*x* + 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

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