TOPICS

Exercise - 1.4

Sets

**Question-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 = Φ

(i) X = {1, 3, 5} Y = {1, 2, 3} X ∪ Y = {1, 2, 3, 5} (ii) A = {a, e, i, o, u} B = {a, b, c} A ∪ B = {a, b, c, e, i, o, u} (iii) A = {x: x is a natural number and multiple of 3} B = {x: x is a natural number less than 6} A = {3, 6, 9, 12,....} and B = {1, 2, 3, 4, 5} A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …} (iv) A = {x: x is a natural number and 1 < x ≤ 6} B = {x: x is a natural number and 6 < x < 10} A = {2, 3, 4, 5, 6} and B = {7, 8, 9} A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9} (v) A = {1, 2, 3}, B = Φ A ∪ B = {1, 2, 3}

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

Here, A = {a, b} and B = {a, b, c} Yes, A ⊂ B. Hence, A ∪ B = {a, b, c} = B

**Question-3 :-** If A and B are two sets such that A ⊂ B, then what is A ∪ B?

Given that A and B are two sets such that A ⊂ B Let A = {a, b} and B = {a, b, c} then A ∪ B = {a, b, c} = B

**Question-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

Given that 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} A ∪ B = {1, 2, 3, 4, 5, 6} (ii) A ∪ C = {1, 2, 3, 4} ∪ {5, 6, 7, 8} A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8} (iii) B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8} B ∪ C = {3, 4, 5, 6, 7, 8} (iv) B ∪ D = {3, 4, 5, 6} ∪ {7, 8, 9, 10} B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10} (v) A ∪ B ∪ C = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {5, 6, 7, 8} A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8} (vi) A ∪ B ∪ D = {1, 2, 3, 4} ∪ {3, 4, 5, 6} ∪ {7, 8, 9, 10} A ∪ B ∪ D = {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} B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

**Question-5 :-** Find the intersection of each pair 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 = Φ

(i) X = {1, 3, 5}, Y = {1, 2, 3} X ∩ Y = {1, 3, 5} ∩ {1, 2, 3} X ∩ Y = {1, 3} (ii) A = {a, e, i, o, u}, B = {a, b, c} A ∩ B = {a, e, i, o, u} ∩ {a, b, c} A ∩ B = {a} (iii) A = {x: x is a natural number and multiple of 3}, B = {x: x is a natural number less than 6} A ∩ B = {3, 6, 9,.....} ∩ {1, 2, 3, 4, 5} A ∩ B = {3} (iv) A = {x: x is a natural number and 1 < x ≤ 6}, B = {x: x is a natural number and 6 < x < 10} A ∩ B = {2, 3, 4, 5, 6} ∩ {7, 8, 9} A ∩ B = Φ (v) A = {1, 2, 3}, B = Φ A ∩ B = {1, 2, 3} ∩ Φ A ∩ B = Φ

**Question-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)

Given that 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} A ∩ B = {7, 9, 11} (ii) B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15} B ∩ C = {7, 11} (iii) A ∩ C ∩ D = {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17} A ∩ C ∩ D = Φ (iv) A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} A ∩ C = {11} (v) B ∩ D = {7, 9, 11, 13} ∩ {15, 17} B ∩ D = Φ (vi) A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}] A ∩ (B ∪ C) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15} A ∩ (B ∪ C) = {7, 9, 11} (vii) A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17} A ∩ D = Φ (viii) A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ [{7, 9, 11, 13} ∪ {15, 17}] A ∩ (B ∪ D) = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13, 15, 17} A ∩ (B ∪ D) = {7, 9, 11} (ix) (A ∩ B) ∩ (B ∪ C) = [{3, 5, 7, 9, 11} ∩ {7, 9, 11, 13}] ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}] (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} (x) (A ∪ D) ∩ (B ∪ C) = [{3, 5, 7, 9, 11} ∪ {15, 17}] ∩ [{7, 9, 11, 13} ∪ {11, 13, 15}] (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 13, 15} (A ∪ D) ∩ (B ∪ C) = {7, 9, 11, 15}

**Question-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 that : A = {x: x is a natural number} = {1, 2, 3,....} B ={x: x is an even natural number} = {2, 4, 6,....} C = {x: x is an odd natural number} = {1, 3, 5,....} D = {x: x is a prime number} = {2, 3, 5,....} (i) A ∩ B = {1, 2, 3,....} ∩ {2, 4, 6,....} A ∩ B = {2, 4, 6,....} i.e., A ∩ B = {x: x is an even natural number} (ii) A ∩ C = {1, 2, 3,....} ∩ {1, 3, 5,....} A ∩ C = {1, 3, 5,....} i.e., A ∩ C = {x: x is an odd natural number} (iii) A ∩ D = {1, 2, 3,....} ∩ {2, 3, 5,....} A ∩ D = {2, 3, 5,....} i.e., A ∩ D = {x: x is a prime number} (iv) B ∩ C = {2, 4, 6,....} ∩ {1, 3, 5,....} B ∩ C = Φ (v) B ∩ D = {2, 4, 6,....} ∩ {2, 3, 5,....} B ∩ D = {2} (vi) C ∩ D = {1, 3, 5,....} ∩ {2, 3, 5,....} C ∩ D = {3, 5, 7,....} i.e., C ∩ D = {x: x is an odd natural number ∉ 1}

**Question-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) A = {1, 2, 3, 4}, B = {x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6} A ∩ B = {1, 2, 3, 4} ∩ {4, 5, 6} A ∩ B = {4} Therefore,{1, 2, 3, 4} and {x: x is a natural number and 4 ≤ x ≤ 6} are not disjoint. (ii) A = {a, e, i, o, u}, B = {c, d, e, f} A ∩ B = {a, e, i, o, u} ∩ (c, d, e, f} A ∩ B = {e} Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint. (iii) A = {x: x is an even integer} ∩ {x: x is an odd integer} = Φ Therefore, this pair of sets is disjoint. {x: x is an even integer} ∩ {x: x is an odd integer} = Φ

**Question-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

Given that 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} A – B = {3, 6, 9, 15, 18, 21} (ii) A – C = {3, 6, 9, 12, 15, 18, 21} - {2, 4, 6, 8, 10, 12, 14, 16} A – C = {3, 9, 15, 18, 21} (iii) A – D = {3, 6, 9, 12, 15, 18, 21} - {5, 10, 15, 20} A – D = {3, 6, 9, 12, 18, 21} (iv) B – A = {4, 8, 12, 16, 20} - {3, 6, 9, 12, 15, 18, 21} B – A = {4, 8, 16, 20} (v) C – A = {2, 4, 6, 8, 10, 12, 14, 16} - {3, 6, 9, 12, 15, 18, 21} C – A = {2, 4, 8, 10, 14, 16} (vi) D – A = {5, 10, 15, 20} - {3, 6, 9, 12, 15, 18, 21} D – A = {5, 10, 20} (vii) B – C = {4, 8, 12, 16, 20} - {2, 4, 6, 8, 10, 12, 14, 16} B – C = {20} (viii) B – D = {4, 8, 12, 16, 20} - {5, 10, 15, 20} B – D = {4, 8, 12, 16} (ix) C – B = {2, 4, 6, 8, 10, 12, 14, 16} - {4, 8, 12, 16, 20} C – B = {2, 6, 10, 14} (x) D – B = {5, 10, 15, 20} - {4, 8, 12, 16, 20} D – B = {5, 10, 15} (xi) C – D = {2, 4, 6, 8, 10, 12, 14, 16} - {5, 10, 15, 20} C – D = {2, 4, 6, 8, 12, 14, 16} (xii) D – C = {5, 10, 15, 20} - {2, 4, 6, 8, 10, 12, 14, 16} D – C = {5, 15, 20}

**Question-10 :-** If X = {a, b, c, d} and Y = {f, b, d, g}, find

(i) X – Y

(ii) Y – X

(iii) X ∩ Y

Given that X = {a, b, c, d} and Y = {f, b, d, g}; (i) X – Y = {a, b, c, d} - {f, b, d, g} X – Y = {a, c} (ii) Y – X = {f, b, d, g} - {a, b, c, d} Y – X = {f, g} (iii) X ∩ Y = {a, b, c, d} ∩ {f, b, d, g} X ∩ Y = {b, d}

**Question-11 :-** If R is the set of real numbers and Q is the set of rational numbers, then what is R – Q?

Given that : R: set of real numbers Q: set of rational numbers Therefore, R – Q is a set of irrational numbers.

**Question-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) A = {2, 3, 4, 5}, B = {3, 6} A ∩ B = {3} A and B are not disjoint. Therefore statement is false. (ii) A = {a, e, i, o, u }, B = {a, b, c, d} A ∩ B = {a} A and B are not disjoint. Therefore statement is false. (iii) A = {2, 6, 10, 14}, B = {3, 7, 11, 15} A ∩ B = Φ A and B are disjoint. Therefore statement is true. (iv) A = {2, 6, 10}, B = {3, 7, 11} A ∩ B = Φ A and B are disjoint. Therefore statement is true.

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