NCERT Solutions class-11 Maths Exercise 1.4

Exercise 1.4

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

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

(ii) A = and B =

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

(iv) A = { is a natural number and } and B = { is a natural number and }

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

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

(ii) A B =

(iii) A B = {1, 2, 3, 4, 5, 6, 9, 12, 15, ………..}

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

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

---

2. Let A = and B = . Is A B? What is A B?

Ans. Given: A = and B = .

Here all elements of set A are present in set B.

A B and A B = = B

---

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

Ans. Given: A and B are two sets such that A B

Taking A = {1, 2} and B = {1, 2, 3}, then A B = {1, 2, 3} = 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

Ans. Given: A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C

= {5, 6, 7, 8} and D = {7, 8, 9, 10}

(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}

= {1, 2, 3, 4, 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 intersections of each pair of sets of question 1 above.

Ans. (i) X Y = {1, 3}

(ii) A B =

(iii) A B = {3, 6, 9, 12, ………..}

(iv) A B =

(v) A B =

---

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) (B C)

(x) (A D) (B C)

Ans. Given: A = {3, 5, 7, 9, 11},

B = {7, 9, 11, 13},

C = {11, 13, 15} and D = {15, 17}

(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} =

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

= {11}

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

(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, 17} = {7, 9, 11}

(vii) A C = {3, 5, 7, 9, 11} {15, 17} =

(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 = { is a natural number}, B = { is an even natural number}, C = { is an odd natural number} and D = { is a prime number}, find:

(i) A B

(ii) A C

(iii) A D

(iv) B C

(v) B D

(vi) C D

Ans. (i) A B = { is a natural number} { is an even natural number} = B

(ii) A C = { is a natural number} { is an odd natural number} = C

(iii) A D = { is a natural number} { is a prime number} = D

(iv) B C = { is an even natural number} { is an odd natural number} =

(v) B D = B C = { is an even natural number} { is a prime number} = {2}

(vi) C D = { is an odd natural number} { is a prime number}

= { is an odd prime number}

---

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

(i) {1, 2, 3, 4} and { is a natural number and }

(ii) and

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

Ans. (i) Let A = {1, 2, 3, 4} and B = { is a natural number and } = {4, 5, 6}

A B = {4}

Therefore, A and B are not disjoint.

(ii) Let A = and B =

A B =

Therefore, A and B are disjoint.

(iii) Let A = { is an even integer} and B = { is an odd integer}

A B =

Therefore, A and B are disjoint.

---

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

Ans. Given: 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};

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

= {3, 6, 9, 15, 18, 21}

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

= {3, 9, 15, 18, 21}

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

= {3, 6, 9, 15, 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, 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, 10, 12 , 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 = and Y = find:

(i) X – Y

(ii) Y – X

(iii) X Y

Ans. Given: X = and Y =

(i) X – Y = =

(ii) Y – X = =

(iii) X Y = =

---

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

Ans. We know that set of real numbers contain rational and irrational numbers.

Therefore, R – Q = set of irrational numbers.

---

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

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

(ii) and 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.

Ans. (i) Let A = {2, 3, 4, 5} and B = {3, 6}

A B = {3}

A and B are not disjoint. Therefore, statement is false.

(ii) Let A = and B =

A B =

A and B are not disjoint. Therefore, statement is false.

(iii) Let A = {2, 6, 10, 14} and B = {3, 7, 11, 15}

A B =

A and B are disjoint. Therefore, statement is true.

(iv) Let A = {2, 6, 10} and B = {3, 7, 11}

A B =

A and B are disjoint. Therefore, statement is true.