# EXERCISE 1.4

**1. Find the union of each of the following pairs of sets:(Ex 1.4 sets ncert maths solution class 11)**

**(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**

**Solution: (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 …}

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 …}

Hence, 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}

Hence, A∪ B = {*x*:* x *∈ N and 1 < *x* < 10}

**(v) **A = {1, 2, 3}, B = Φ

A∪ B = {1, 2, 3}

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

**Solution: **It is given that

A = {*a*, *b*} and B = {*a*, *b*, *c*}

Yes, A ⊂ B

**So, A∪ B = { a, b, c} = B**

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

**Solution: **If A and B are two sets such that A ⊂ B,

then 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**

**Solution: **It is 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, 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}

**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**

**Solution:** **(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

**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)**

**Solution: (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}

**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**

**Solution: **It can be written as

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}

={2, 4, 6, 8…} = B

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

={1, 3, 5, 7….} = C

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

={2, 3, 5, 7…} = D

**(iv)** B ∩ C = Φ

**(v) **B ∩ D = {2}

**(vi) **C ∩ D = {*x*:* x* is 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}**

**Solution: (i)** {1, 2, 3, 4}

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

So we get

{1, 2, 3, 4} ∩ {4, 5, 6} = {4}

Hence, this pair of sets is not disjoint.

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

Hence, {*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

Hence, this pair of sets is 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**

**Solution: ** **(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}

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

**(i) X – Y**

**(ii) Y – X**

**(iii) X ∩ Y**

**Solution: (i) **X – Y = {*a*, *c*}

**(ii) **Y – X = {*f*, *g*}

**(iii)** X ∩ Y = {*b*, *d*}

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

**Solution: **We know that

R – Set of real numbers

Q – Set of rational numbers

Hence, R – Q is a 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.
**

**Solution: (i) False**

If 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}

So we get {2, 3, 4, 5} ∩ {3, 6} = {3}

**(ii) False**

If *a* ∈ {*a*, *e*, *i*, *o*, *u*}, *a* ∈ {*a*, *b*, *c*, *d*}

So we get {*a*, *e*, *i*, *o*, *u*} ∩ {*a*, *b*, *c*, *d*} = {*a*}

**(iii) True**

Here {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ

**(iv) True**

Here {2, 6, 10} ∩ {3, 7, 11} = Φ