11th NCERT Sets Exercise 1.4 Questions 12

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Question (1)

Find the union of each of the following pairs of sets:(i) X = {1, 3, 5} Y = {1, 2, 3}

Solution

X ∪ Y = { 1, 2, 3, 5 }(ii) A = {a, e, i, o, u} B = {a, b, c}

Solution

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}

Solution

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

A ∪ B = { 1, 2, 3, 4, 5, 6, 9, 12, 15, ....... }

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

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

Solution

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

A ∪ B = { 2, 3, 4, 5, 6, 7, 8, 9}

(v) A = {1, 2, 3}, B = Φ

Solution

A ∪ B = { 1, 2, 3} = A.Question (2)

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

As all elements of set A belongs to set B.Yes A ⊂ B.A ∪ B = { a, b, c} = B

Question (3)

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

A and B are two sets such that A ⊂ B.So all elements of set A are belongs to set B.

So A ∪ B = 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

Solution

A ∪ B = { 1, 2, 3, 4, 5, 6}(ii) A ∪ C

Solution

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

Solution

B ∪ C = { 3, 4, 5, 6, 7, 8}(iv) B ∪ D

Solution

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

Solution

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

Solution

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

Solution

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}

Solution

X ∩ Y = { 1, 3 }(ii) A = {a, e, i, o, u} B = {a, b, c}

Solution

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}

Solution

A = { 3, 6, 9, 12, .....}B = { 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}

Solution

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

A ∩ B = { }

(v) A = {1, 2, 3}, B = Φ

Solution

A ∩ B = { }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

Solution

A ∩ B = { 7, 9, 11}(ii) B ∩ C

Solution

B ∩ C = { 11, 13}(iii) A ∩ C ∩ D

Solution

A ∩ C ∩ D = {}(iv) A ∩ C

Solution

A ∩ C = {11}(v) B ∩ D

Solution

B ∩ D = { }(vi) A ∩ (B ∪ C)

Solution

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

(vii) A ∩ D

Solution

A ∩ D = { }(viii) A ∩ (B ∪ D)

Solution

B ∪ D = { 7, 9, 11, 13, 15, 17}A ∩ (B ∪D ) = { 7, 9, 11 }

(ix) (A ∩ B) ∩ (B ∪ C)

Solution

A ∩ B = { 7, 9 , 11}B ∪ C = {7,9,11, 13, 15}

( A ∩ B) ∩ (B ∪ C) = {7, 9, 11 }

(x) (A ∪ D) ∩ (B ∪ C)

Solution

A ∪ D = {3, 5, 7, 9, 11, 15, 17}B ∪ C = {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}, findSolution

A = 1, 2, 3, 4, ........} , B = { 2, 4, 6, 8 .....}C = { 1, 3, 5, 7, 9, ....}, D = { 2, 3, 5, 7, 11, 13, ....} (i) A ∩ B

Solution

A ∩ B = { 2, 4, 6, 8, ....} = B(ii) A ∩ C

Solution

A ∩ C = { 1, 3, 5, 7, ....} = C.(iii) A ∩ D

Solution

A ∩ D = { 2, 3, 5, 7, 11, 13, .....} = D.(iv) B ∩ C

Solution

B = ∩ C = {}(v) B ∩ D

Solution

B ∩ D = {2}(vi) C ∩ D

Solution

C ∩ D = { 3, 5, 7, 11, ....} = D - {2}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}

Solution

A = { 1, 2, 3, 4} and B = { 4, 5. 6}A ∩ B = { 4}

It is not empty set, so not disjoint.

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

Solution

A = { a, e, i, o, u} and B = { c, d, e, f}A ∩ B = {e}

It is not empty set, so not disjoint.

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

Solution

A = {2, 4, 6, 8, ....}B = { 1, 3, 5, 7, ....}

A ∩ B = {}

It is empty set, so disjoint.

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

Solution

A ∩B = { {12}A - B = { x; x ∈A, x ∉ B }

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

(ii) A – C

Solution

A ∩ C = { 6, 12}A - C = { x; x ∈A, x ∉ C }

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

(iii) A – D

Solution

A ∩ D = { 15}A - D = { x; x ∈A, x ∉D }

= { 3, 6, 9, 12, 18, 21}

(iv) B – A

Solution

A ∩B = { {12}B - A = { x; x ∈ B, x ∉ A}

= { 4, 8, 16, 20}

(v) C – A

Solution

A ∩ C = { 6, 12}C - A = { x; x ∈ C, x ∉ A }

= { 2, 4, 8, 10, 14, 16}

(vi) D – A

Solution

A ∩ D = { 15}D - A = { x; x ∈D, x ∉A}

= { 5, 10, 20}

(vii) B – C

Solution

B ∩ C = { 4, 8, 12, 16}B - C = { x : x ∈ B, x ∉ C}

= { 20}

(viii) B – D

Solution

B ∩ D = { 20}B - D = { x : x ∈ B, x ∉ D}

= { 4, 8, 12, 16}

(ix) C – B

Solution

B ∩ C = { 4, 8, 12, 16}C - B = { x : x ∈ C, x ∉ B}

= { 2, 6, 10, 14}

(x) D – B

Solution

B ∩ D = { 20}D - B = { x : x ∈ D, x ∉ B}

= { 5, 10, 15}

(xi) C – D

Solution

C ∩ D = { {10}C - D = {x : x ∈ C, x ∉ D }

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

(xii) D – C

Solution

C ∩ D = { {10}D - C = {x : x ∈ D, x ∉ C }

= { 5, 15, 20}

Question (10)

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

Solution

X ∩ Y = { b ,d}X - Y = {x : x ∈ X x ∉ Y }

= { a, c}

(ii) Y – X

Solution

X ∩ Y = { b ,d}Y - X = {x : x ∈Y, x ∉ X }

= {f, g}

(iii) X ∩ Y

Solution

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?Solution

R ∩ Q = QR -Q = {x : x ∈R, x ∉ Q }

= set of irrational numbers.= Q'

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.

Solution

Let A = { 2, 3, 4, 5} and B = { 3, 6}A ∩ B = {3}

As it is not null set, they are not disjoint set.

False, they are not disjoint sets.

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

Solution

Let A = {a, e, i, o, u } and B = {a, b, c, d}A ∩ B = {a}

As it is not null set, they are not disjoint set.

False, they are not disjoint sets.

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

Solution

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

As it is null set, they are disjoint set.

True, they are disjoint sets.

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

Solution

Let A = {2, 6, 10} and B = {3, 7, 11}A ∩ B = {}

As it is null set, they are disjoint set.

True they are disjoint sets.