**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

**Solution:**

(i)X = {1, 3, 5} Y = {1, 2, 3}So, the union of the pairs of set can be written as

X ∪ Y= {1, 2, 3, 5}

(ii)A = {a, e, i, o, u} B = {a, b, c}So, the union of the pairs of set can be written as

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}

So, the union of the pairs of set can be written as

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}

So, the union of the pairs of set can be written as

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 = ΦSo, the union of the pairs of set can be written as

A∪ B = {1, 2, 3}

**Question 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, the union of the pairs of set can be written as

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:**

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

**(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}

### 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

**Solution:**

(i)X = {1, 3, 5}, Y = {1, 2, 3}So, the intersection of the given set can be written as

X ∩ Y = {1, 3}

(ii)A = {a, e, i, o, u}, B = {a, b, c}So, the intersection of the given set can be written as

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}

So, the intersection of the given set can be written as

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}

So, the intersection of the given set can be written as

A ∩ B = Φ

(v)A = {1, 2, 3}, B = ΦSo, the intersection of the given set can be written as

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)**

**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}

**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**

**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} = 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 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} = ΦHence, this pair of sets is 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**

**(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}

**Question 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}

**Question 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.

**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.**

**Solution:**

(i)FalseIf 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}

So, we get {2, 3, 4, 5} ∩ {3, 6} = {3}

(ii)FalseIf 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)TrueHere {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ

(iv)TrueHere {2, 6, 10} ∩ {3, 7, 11} = Φ