## NCERT Solutions for Class 11 Maths Chapter 1 Sets Exercise 1.4

Chapter 1 Sets Exercise 1.4 NCERT Solutions for Class 11 Maths will help you in solving difficult questions and completing your homework on time. Class 11 Maths NCERT Solutions will improve your problem solving skills and marks in the exams. It will also help you in preparing for higher classes and going for advance books.

1. Find the union of each of the following pairs of sets:

(i) X = {1, 3, 5}, Y = {1, 2, 3}

(ii) A = {a, e, i, o, u}, B = {a, b, c}

(iii) A = {x: x is a natural number and multiple of 3} B = {x: x is a natural number less than 6}.

(iv) A = {x: x is a natural number and 1 < x ≤ 6 }

B = {x: x is a natural number and 6 < x < 10}

(v) A = {1, 2, 3}, B = Ð¤

**Answer**

(i) X ∪ Y = {1, 3, 5}∪ {1, 2, 3}

= {1, 2, 3, 5}

(ii) A ∪ B = {a, e, i, o, u} ∪ {a, b, c}

= {a, b, c, e, i, o, u}

(iii) A = {3, 6, 9, ....}

B = {1, 2, 3, 4, 5}

Hence,

A ∪B = {3, 6, 9, 12,...} ∪ {1, 2, 3, 4, 5}

= {1, 2, 3, 4, 5, 6, 9, 12, ....}

= {x: x = 1 = 1, 2, 3, 4, 5 or a multiple of 3}

(iv) A = {2, 3, 4, 5, 6} B = {7, 8, 9}

∴ A ∪B = {2, 3, 4, 5, 6, 7, 8, 9}

= {x: 1 < x < 10, x Îµ N}

(v) A ∪ B = {1, 2, 3} ∪ {}= {1, 2, 3}.

2. Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

Yes, A ⊂ B. Because, every element of A is also an element of B, therefore, A is a subset of B.

2. Let A = {a, b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?

**Answer**Yes, A ⊂ B. Because, every element of A is also an element of B, therefore, A is a subset of B.

A ∪ B = {a, b}∪(a, b, c) = {a, b, c}

3. If A and B are two sets such that A⊂ B, then what is A ∪ B?

Since, A is a subset of B, therefore, every element of set A is contained in the set B. Hence, A ∪ B = B.

4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

(i) A ∪ B

(ii) A ∪ C

(iii) B ∪ C

(iv) B ∪ D

(v) A ∪ B ∪ C

(vi) A ∪ B∪ D

(vii) B ∪ C ∪ D

(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

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

(iii) B ∪ C = {3, 4, 5, 6}∪ {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}

(iv) B ∪D = {3, 4, 5, 6}∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C = {1, 2, 3, 4}∪ {3, 4, 5, 6}∪ {5, 6, 7, 8}

(vi) A ∪ B ∪ D = {1, 2, 3, 4}∪ {3, 4, 5, 6} ∪ {7, 8, 9, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B ∪ C ∪ D = {3, 4, 5, 6}∪ {5, 6, 7, 8} ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}

5. Find the intersection of each pair of sets of Q. 1, above.

(i) X ∩Y= {1, 3, 5}∩ {1, 2, 3} = {1, 3}

(ii) A ∩ B = {a, e, i, o, u}∩ {a, b, c}= {a}

(iii) A ∩ B = {3, 6, 9, ....} ∩ {1, 2, 3, 4, 5}= {3}

(iv) A ∩ B = {2, 3, 4, 5, 6}∩ {7,8,9}= f

(v) A ∩ B = {1, 2, 3} ∩ Ð¤ = Ð¤

6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

(i) A ∩ B

(ii) B ∩ C

(iii) A ∩ C ∩ D

(iv) A ∩ C

(v) B ∩ D

(vi) A ∩ (B ∪ C)

(vii) A ∩ D

(viii) A ∩ (B ∪ D)

(ix) (A ∩ B) Ã‡ (B ∪ C)

(x) (A∪ D) ∩ (B ∪ C)

(i) A ∩ B = {3, 5, 7, 9, 11}∩ {7, 9, 11, 13}

= {7, 9, 11}

(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15}

= {11, 13}

(iii) A ∩ C ∩ D

= {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17}

={11} ∩ {15, 17} = Ð¤

(iv) A∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}

(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = Ð¤

(vi) A ∩ (B ∪ C)

= {3, 5, 7, 9, 11}∩ [{7, 9, 11, 13}

∪ {11, 13, 15}]

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

(vii) A ∩ D = {3, 5, 7, 9, 11}∩ {15, 17}= Ð¤

(viii) A∩ (B∪D)

= {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {15, 17}]

= {3, 5, 7, 9, 11}∩ {7, 9, 11, 13, 15, 17}

= {7, 9, 11}

(ix) (A ∩ B) ∩ (B ∪ C)

= [{3, 5, 7, 9, 11}∩ {7, 9, 11, 13}]

∩ [{7, 9, 11, 13}∪ {11, 13, 15}]

= {7, 9, 11,}∩ {7, 9,11,13,15}

= {7, 9, 11}.

(x) (A∪ D) ∩ (B ∪ C)

= [{3, 5, 7, 9, 11}∪ {15, 17}]

∩ [{7, 9, 11, 13}∪ {11, 13, 15}]

= {3, 5, 7, 9, 11, 15, 17}∩ {7, 9, 11, 13, 15}

= {7, 9, 11, 15}.

7. If A = {x: x is a natural number},

B = {x: x is an even natural number}

C = {x: x is an odd natural number} and

D = {x: x is a prime number}, find

(i) A ∩ B

(ii) A ∩ C

(iii) A ∩ D

(iv) B ∩ C

(v) B ∩ D

(vi) C ∩ D

Given, A = {1, 2, 3, 4, .....}

B = {2, 4, 6, 8, .....}

C = {1, 3, 5, 7, ....}

and D = {2, 3, 5, 7, 11, 13, ....}

(i) A ∩B= {1, 2, 3, 4,...} ∩ {2, 4, 6, 8,...}

= {2, 4, 6, 8, ....} = B

(ii) A ∩C = {1, 2, 3, 4, ...}∩ {1, 3, 5, 7}

= {1, 3, 5, 7, ....}= C

(iii) A∩D = {1, 2, 3, 4, ....}∩ {2, 3, 5, 7, 11, 13, ....}

= {2, 3, 5, 7, 11, 13,...} = D

(iv) B ∩C = {2, 4, 6, 8, ...}∩ {1, 3, 5, 7,...} = Ð¤

(v) B ∩ D = {2, 4, 6, 8,...} ∩ {2, 3, 5, 7, 11, 13,

....}= {2}

(vi) C∩D = {1, 3, 5, 7,...} ∩ {2, 3, 5, 7, 11, 13, ....}

= {3, 5, 7, 11, 13, ....}

= {x: x is an odd prime number}.

8. Which of the following pairs of sets are disjoint:

(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x≤6}.

(ii) {a, e, i,o, u} and {c, d, e, f}

(iii) {x: x is an even integer} and {x: x is an odd integer}

(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6 }i.e. {4, 5, 6} are not disjoint sets as they have 4 as a common element.

(ii) {a, e, i,o, u} and {c, d, e, f} are not disjoint sets, because they have common element e.

(iii) {x: x is an even integer} = {+ 2, + 4, + 6,...} and {x: x is an odd integer} = {± 1, ± 3, ± 5,...} are disjoint sets, because they have no common element.

9. If A = {3, 6, 9, 12, 15, 18, 21},

B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12, 14, 16},

D = {5, 10, 15, 20}, find

(i) A – B

(ii) A – C

(iii) A – D

(iv) B – A

(v) C – A

(vi) D – A

(vii) B – C

(viii) B – D

(ix) C – B

(x) D – B

(xi) C – D

(xii) D – C

(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}= {3, 6, 9, 15, 18, 21}

(ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}

(iii) A – D = {{3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20} = {3, 6, 9, 12, 18, 21}.

(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}.

(v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}

(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}

(vii) B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16}= {20}

(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20} = {4, 8, 12, 16}.

(ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20} = {2, 6, 10, 14}.

(x) D – B = {5, 10, 15, 20}– {4, 8, 12, 16, 20} = {5, 10, 15}

(xi) C – D = {2, 4, 6, 8, 12, 10, 14, 16} – {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}

(xii) D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {5, 15, 20}

10. If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y

(ii) Y – X

(iii) X ∩ Y

(i) X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}

(ii) Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}

(iii) X ∩ Y.= {a, b, c, d} ∩ {f, b, d, g}= {b, d}

11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

R – Q = {set of real numbers} – {set of rational numbers}

= {set of rational numbers and set of irrational numbers}– {set of rational numbers}

= set of irrational numbers

12. State whether each of the following statement is true or false. Justify your answer:

(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.

(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.

(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets

(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets

(i) False.

{2, 3, 4, 5} ∩ {3, 6} = {3} ≠ Ð¤

The given sets are not disjoint.

(ii) False.

{a, e, i, o, u}∩ {a, b, c, d}= {a} ≠ Ð¤

The given sets are not disjoint.

(iii) True.

{2, 6, 10, 14}∩ {3, 7, 11, 15} ≠ Ð¤

The given sets are disjoint.

(iv) True.

{2, 6, 10} ∩ {3, 7, 11} ≠ Ð¤

The given sets are disjoint.

3. If A and B are two sets such that A⊂ B, then what is A ∪ B?

**Answer**Since, A is a subset of B, therefore, every element of set A is contained in the set B. Hence, A ∪ B = B.

4. If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find

(i) A ∪ B

(ii) A ∪ C

(iii) B ∪ C

(iv) B ∪ D

(v) A ∪ B ∪ C

(vi) A ∪ B∪ D

(vii) B ∪ C ∪ D

**Answer**(i) A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

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

(iii) B ∪ C = {3, 4, 5, 6}∪ {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}

(iv) B ∪D = {3, 4, 5, 6}∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C = {1, 2, 3, 4}∪ {3, 4, 5, 6}∪ {5, 6, 7, 8}

(vi) A ∪ B ∪ D = {1, 2, 3, 4}∪ {3, 4, 5, 6} ∪ {7, 8, 9, 10} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B ∪ C ∪ D = {3, 4, 5, 6}∪ {5, 6, 7, 8} ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}

5. Find the intersection of each pair of sets of Q. 1, above.

**Answer**(i) X ∩Y= {1, 3, 5}∩ {1, 2, 3} = {1, 3}

(ii) A ∩ B = {a, e, i, o, u}∩ {a, b, c}= {a}

(iii) A ∩ B = {3, 6, 9, ....} ∩ {1, 2, 3, 4, 5}= {3}

(iv) A ∩ B = {2, 3, 4, 5, 6}∩ {7,8,9}= f

(v) A ∩ B = {1, 2, 3} ∩ Ð¤ = Ð¤

6. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}; find

(i) A ∩ B

(ii) B ∩ C

(iii) A ∩ C ∩ D

(iv) A ∩ C

(v) B ∩ D

(vi) A ∩ (B ∪ C)

(vii) A ∩ D

(viii) A ∩ (B ∪ D)

(ix) (A ∩ B) Ã‡ (B ∪ C)

(x) (A∪ D) ∩ (B ∪ C)

**Answer**(i) A ∩ B = {3, 5, 7, 9, 11}∩ {7, 9, 11, 13}

= {7, 9, 11}

(ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15}

= {11, 13}

(iii) A ∩ C ∩ D

= {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17}

={11} ∩ {15, 17} = Ð¤

(iv) A∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}

(v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = Ð¤

(vi) A ∩ (B ∪ C)

= {3, 5, 7, 9, 11}∩ [{7, 9, 11, 13}

∪ {11, 13, 15}]

= {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

(vii) A ∩ D = {3, 5, 7, 9, 11}∩ {15, 17}= Ð¤

(viii) A∩ (B∪D)

= {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {15, 17}]

= {3, 5, 7, 9, 11}∩ {7, 9, 11, 13, 15, 17}

= {7, 9, 11}

(ix) (A ∩ B) ∩ (B ∪ C)

= [{3, 5, 7, 9, 11}∩ {7, 9, 11, 13}]

∩ [{7, 9, 11, 13}∪ {11, 13, 15}]

= {7, 9, 11,}∩ {7, 9,11,13,15}

= {7, 9, 11}.

(x) (A∪ D) ∩ (B ∪ C)

= [{3, 5, 7, 9, 11}∪ {15, 17}]

∩ [{7, 9, 11, 13}∪ {11, 13, 15}]

= {3, 5, 7, 9, 11, 15, 17}∩ {7, 9, 11, 13, 15}

= {7, 9, 11, 15}.

7. If A = {x: x is a natural number},

B = {x: x is an even natural number}

C = {x: x is an odd natural number} and

D = {x: x is a prime number}, find

(i) A ∩ B

(ii) A ∩ C

(iii) A ∩ D

(iv) B ∩ C

(v) B ∩ D

(vi) C ∩ D

**Answer**Given, A = {1, 2, 3, 4, .....}

B = {2, 4, 6, 8, .....}

C = {1, 3, 5, 7, ....}

and D = {2, 3, 5, 7, 11, 13, ....}

(i) A ∩B= {1, 2, 3, 4,...} ∩ {2, 4, 6, 8,...}

= {2, 4, 6, 8, ....} = B

(ii) A ∩C = {1, 2, 3, 4, ...}∩ {1, 3, 5, 7}

= {1, 3, 5, 7, ....}= C

(iii) A∩D = {1, 2, 3, 4, ....}∩ {2, 3, 5, 7, 11, 13, ....}

= {2, 3, 5, 7, 11, 13,...} = D

(iv) B ∩C = {2, 4, 6, 8, ...}∩ {1, 3, 5, 7,...} = Ð¤

(v) B ∩ D = {2, 4, 6, 8,...} ∩ {2, 3, 5, 7, 11, 13,

....}= {2}

(vi) C∩D = {1, 3, 5, 7,...} ∩ {2, 3, 5, 7, 11, 13, ....}

= {3, 5, 7, 11, 13, ....}

= {x: x is an odd prime number}.

8. Which of the following pairs of sets are disjoint:

(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x≤6}.

(ii) {a, e, i,o, u} and {c, d, e, f}

(iii) {x: x is an even integer} and {x: x is an odd integer}

**Answer**(i) {1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6 }i.e. {4, 5, 6} are not disjoint sets as they have 4 as a common element.

(ii) {a, e, i,o, u} and {c, d, e, f} are not disjoint sets, because they have common element e.

(iii) {x: x is an even integer} = {+ 2, + 4, + 6,...} and {x: x is an odd integer} = {± 1, ± 3, ± 5,...} are disjoint sets, because they have no common element.

9. If A = {3, 6, 9, 12, 15, 18, 21},

B = {4, 8, 12, 16, 20},

C = {2, 4, 6, 8, 10, 12, 14, 16},

D = {5, 10, 15, 20}, find

(i) A – B

(ii) A – C

(iii) A – D

(iv) B – A

(v) C – A

(vi) D – A

(vii) B – C

(viii) B – D

(ix) C – B

(x) D – B

(xi) C – D

(xii) D – C

**Answer**(i) A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}= {3, 6, 9, 15, 18, 21}

(ii) A – C = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12, 14, 16} = {3, 9, 15, 18, 21}

(iii) A – D = {{3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20} = {3, 6, 9, 12, 18, 21}.

(iv) B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21} = {4, 8, 16, 20}.

(v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} – {3, 6, 9, 12, 15, 18, 21} = {2, 4, 8, 10, 14, 16}

(vi) D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21} = {5, 10, 20}

(vii) B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12, 14, 16}= {20}

(viii) B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20} = {4, 8, 12, 16}.

(ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} – {4, 8, 12, 16, 20} = {2, 6, 10, 14}.

(x) D – B = {5, 10, 15, 20}– {4, 8, 12, 16, 20} = {5, 10, 15}

(xi) C – D = {2, 4, 6, 8, 12, 10, 14, 16} – {5, 10, 15, 20} = {2, 4, 6, 8, 12, 14, 16}

(xii) D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12, 14, 16} = {5, 15, 20}

10. If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y

(ii) Y – X

(iii) X ∩ Y

**Answer**(i) X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}

(ii) Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}

(iii) X ∩ Y.= {a, b, c, d} ∩ {f, b, d, g}= {b, d}

11. If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

**Answer**R – Q = {set of real numbers} – {set of rational numbers}

= {set of rational numbers and set of irrational numbers}– {set of rational numbers}

= set of irrational numbers

12. State whether each of the following statement is true or false. Justify your answer:

(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.

(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.

(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets

(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets

**Answer**(i) False.

{2, 3, 4, 5} ∩ {3, 6} = {3} ≠ Ð¤

The given sets are not disjoint.

(ii) False.

{a, e, i, o, u}∩ {a, b, c, d}= {a} ≠ Ð¤

The given sets are not disjoint.

(iii) True.

{2, 6, 10, 14}∩ {3, 7, 11, 15} ≠ Ð¤

The given sets are disjoint.

(iv) True.

{2, 6, 10} ∩ {3, 7, 11} ≠ Ð¤

The given sets are disjoint.