CBSE Class XI

Question 1:

If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩Y).

Answer

It is given that:

n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38
n(X ∩ Y) = ?

We know that:

n(X ∪ Y) = n(X) + n(Y) – n(X ∩ Y)

∴ 38 = 17 + 23 – n(X∩Y)

⇒n(X ∩ y) = 40 – 38 = 2

∴ n (X ∩ Y) = 2

Question 7:

Fill in the blanks to make each of the following a true statement:

(i) A ∪A΄ = …

(ii) Φ′ ∩ A = …

(iii) A ∩ A΄ = …

(iv) U΄ ∩ A = …

Answer

(i) A ∪ A΄ = U

(ii) Φ′ ∩ A = U ∩ A = A

∴ Φ′ ∩ A = A

(iii) A ∩ A′ = Φ

(iv) U′ ∩ A = Φ ∩ A = Φ

∴ U′ ∩ A = Φ

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

A΄ is the set of all equilateral triangles.

Question 5:

Draw appropriate Venn diagram for each of the following:

(i) (A ∪ B)΄          

(ii) A΄∩ B΄           

(iii) (A ∩ B)΄

(iv) A΄ ∪ B΄

Answer

(i) (A ∪ B)΄

(ii) A΄∩ B΄

 

(iii) (A ∩ B)΄

 

(iv) A΄ ∪ B΄

 

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΄    (ii) (A ∩B)΄ = A΄∪ B΄

Answer

U = {1, 2, 3, 4, 5, 6, 7, 8, 9}

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

(i) (A ∪ B)΄ = {2, 3, 4, 5, 6, 7, 8}΄ = {1, 9}

∴(A ∪ B)΄ = A΄∩ B΄

(ii) (A ∩ B)΄ = {2}΄ ={1, 3, 4, 5, 6, 7, 8, 9}

A΄ ∪ B΄ ={1, 3, 5, 7, 9} ∪ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}

∴(A ∩ B)΄ = A΄ ∪ B΄

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}

(ii) {x: x is an odd natural number} 

(iii) {x: x is a positive multiple of 3}

(iv){x: x is a prime number}

(v){x: x is a natural number divisible by 3 and 5}

(vi) {x: x is a perfect square}

(vii) {x: x is perfect cube}

(viii) {x: x + 5 = 8}

(ix) {x: 2x + 5 = 9}

(x) {x: x ≥ 7}

(xi) {x: x ∈ N and 2x + 1 > 10}

Answer

U = N: Set of natural numbers

(i) {x: x is an even natural number}´ = {x: x is an odd natural number}

(ii) {x: x is an odd natural number}´ = {x: x is an even natural number}

(iii) {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}´ ={x: x is a positive composite number and x = 1}

(v) {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}´ = {x: x ∈ N and x is not a perfect square}

(vii) {x: x is a perfect cube}´ = {x: x ∈ N and x is not a perfect cube}

(viii) {x: x + 5 = 8}´ = {x: x ∈ N and x ≠ 3}

(ix) {x: 2x + 5 = 9}´ = {x: x ∈ N and x ≠ 2}

(x) {x: x ≥ 7}´ = {x: x ∈ N and x < 7}

(xi) {x: x ∈ N and 2x + 1 > 10}´ = {x: x ∈ N and x ≤ 9/2}

Question 2:

If U = {a, b, c, d, e, f, g, h}, find the complements of the following sets:

(i) A = {a, b, c}

(ii) B = {d, e, f, g}

(iii) C = {a, c, e, g}

(iv) D = {f, g, h, a}

Answer

U = {a, b, c, d, e, f, g, h}

(i) A = {a, b, c}

A΄ = {d ,e f, g, h}

(ii) B = {d, e, f, g}

∴ B΄ = {a, b, c, h}

(iii) C = {a, c, e, g}

∴ C΄ = {b, d, f, h}

(iv) D = {f, g, h, a}

∴ D΄={b, c, d, e}

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΄

(ii) B΄

(iii) (A ∪ C)΄

(iv) (A ∪ B)΄

(v) (A΄)΄

(vi)(B−C)΄

Answer

U ={1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4}

B = {2, 4, 6, 8}
C = {3, 4, 5, 6}

(i) A΄={5, 6, 7, 8, 9}

(ii) B΄ = {1, 3, 5, 7, 9}

(iii) A ∪ C={1, 2, 3, 5, 6, 8}

∴ (A ∪ C)΄ = {7, 8, 9}

(iv) A ∪ B = {1, 2, 3, 4, 6, 8}

(A ∪ B)΄ = {5, 7, 9}

(v) (A΄)΄ = A ={1, 2, 3, 4}

 (vi) B – C = {2, 8}

∴ (B−C)΄ = {1, 3, 4, 5, 6, 7, 9}

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.

Answer

(i) False

As 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}
⇒ {2, 3, 4, 5} ∩ {3, 6} = {3}
(ii) False

As a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}
⇒ {a, e, i, o, u } ∩ {a, b, c, d} = {a}

(iii) True

As {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ
(iv) True

As {2, 6, 10} ∩ {3, 7, 11} = Φ

Question 11:

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

Answer

R: set of real numbers

Q: set of rational numbers

Therefore, R – Q is a set of irrational numbers.

Question 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, c}
(ii) Y – X = {f, g}
(iii) X ∩ Y = {b, d}

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

Answer

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

(v) C – A = {2, 4, 8, 10, 14, 16}
(vi) D – A = {5, 10, 20}
(vii) B – C = {20}

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

(x) D – B = {5, 10, 15}

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

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}

Answer

(i) {1, 2, 3, 4}

{x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

Now, {1, 2, 3, 4} ∩ {4, 5, 6} = {4}

Therefore, this pair of sets is not disjoint.

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

Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.

(iii) {x: x is an even integer} ∩ {x: x is an odd integer} = Φ Therefore, this pair of sets is disjoint.

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

Answer

A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}

B ={x: x is an even natural number} = {2, 4, 6, 8 …}

C = {x: x is an odd natural number} = {1, 3, 5, 7, 9…} D = {x: x is a prime number} = {2, 3, 5, 7 …}

(i) A ∩B = {x: x is a even natural number} = B

(ii) A ∩ C = {x: x is an odd natural number} = C

(iii) A ∩ D = {x: x is a prime number} = D

(iv) B ∩ C = Φ

(v) B ∩ D = {2}

(vi) C ∩ D = {x: x is odd prime number}

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)

Answer

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

(iii) A ∩ C ∩ D = { A ∩ C} ∩ D = {11} ∩ {15, 17} = Φ

(iv) A ∩ C = {11}

(v) B ∩ D = Φ

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

(vii) A ∩ D = Φ

(viii) A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D) = = {7, 9, 11} ∪ Φ = {7, 9, 11}

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

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

Answer

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

X ∩ Y = {1, 3}

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

(iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}

B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}

∴ A ∩ B = {3}

(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

A ∩ B = Φ

(v) A = {1, 2, 3}, B = Φ

A ∩ B = Φ

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

Answer

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, 5, 6}

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

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

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

Question 3 :

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

Answer

If A and B are two sets such that A ⊂ B, then A ∪ B = B

Question 2 :

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

Answer

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

A∪ B = {a, b, c} = B

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

Answer

(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} = {3, 6, 9 …} As B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6} A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}

∴ A ∪ B = {x: x = 1, 2, 4, 5 or a multiple of 3}

(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6} B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

A∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

∴ A∪ B = {x: x ∈ N and 1 < x < 10}

(v) A = {1, 2, 3}, B = Φ

A∪ B = {1, 2, 3}