# Class 11 NCERT Solutions – Chapter 1 Sets – Exercise 1.4

**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 **â‰¤ 6}

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

### (v) A = {1, 2, 3}, B = Î¦

**Solution:**

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

X âˆª 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

Aâˆª B = {a, b, c, e, i, o, u}

(iii)A = {x: x is a natural number and multiple of 3} = {3, 6, 9 â€¦}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 âˆª B = {1, 2, 4, 5, 3, 6, 9, 12 â€¦}

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

(iv)A = {x: x is a natural number and 1 < x â‰¤ 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

Aâˆª B = {2, 3, 4, 5, 6, 7, 8, 9}

Hence, Aâˆª B = {x: x âˆˆ N and 1 < x < 10}

(v)A = {1, 2, 3}, B = Î¦So, the union of the pairs of set can be written as

Aâˆª B = {1, 2, 3}

**Question 2. Let A = {a, b}, B = {a, b, c}. Is A âŠ‚ B? What is A âˆª B?**

**Solution:**

It is given that

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

Yes, A âŠ‚ B

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

Aâˆª B = {a, b, c} = B

**Question 3. If A and B are two sets such that A âŠ‚ B, then what is A âˆª B?**

**Solution:**

If A and B are two sets such that A âŠ‚ B, then A âˆª 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 âˆª B**

**(ii) A âˆª C**

**(iii) B âˆª C**

**(iv) B âˆª D**

**(v) A âˆª B âˆª C**

**(vi) A âˆª B âˆª D**

**(vii) B âˆª C âˆª 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 âˆª B = {1, 2, 3, 4, 5, 6}

(ii)A âˆª C = {1, 2, 3, 4, 5, 6, 7, 8}

(iii)B âˆª C = {3, 4, 5, 6, 7, 8}

(iv)B âˆª D = {3, 4, 5, 6, 7, 8, 9, 10}

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

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

(vii)B âˆª C âˆª 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 â‰¤ 6}

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

### (v) A = {1, 2, 3}, B = Î¦

**Solution:**

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

X âˆ© 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 âˆ© B = {a}

(iii)A = {x: x is a natural number and multiple of 3} = (3, 6, 9 â€¦}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 âˆ© B = {3}

(iv)A = {x: x is a natural number and 1 < x â‰¤ 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 âˆ© B = Î¦

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

A âˆ© B = Î¦

### 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 âˆ© B**

**(ii) B âˆ© C**

**(iii) A âˆ© C âˆ© D**

**(iv) A âˆ© C**

**(v) B âˆ© D**

**(vi) A âˆ© (B âˆª C)**

**(vii) A âˆ© D**

**(viii) A âˆ© (B âˆª D)**

**(ix) (A âˆ© B) âˆ© (B âˆª C)**

**(x) (A âˆª D) âˆ© (B âˆª C)**

**Solution:**

(i)A âˆ© B = {7, 9, 11}

(ii)B âˆ© C = {11, 13}

(iii)A âˆ© C âˆ© D = {A âˆ© C} âˆ© D= {11} âˆ© {15, 17}

= Î¦

(iv)A âˆ© C = {11}

(v)B âˆ© D = Î¦

(vi)A âˆ© (B âˆª C) = (A âˆ© B) âˆª (A âˆ© C)= {7, 9, 11} âˆª {11}

= {7, 9, 11}

(vii)A âˆ© D = Î¦

(viii)A âˆ© (B âˆª D) = (A âˆ© B) âˆª (A âˆ© D)= {7, 9, 11} âˆª Î¦

= {7, 9, 11}

(ix)(A âˆ© B) âˆ© (B âˆª C) = {7, 9, 11} âˆ© {7, 9, 11, 13, 15}= {7, 9, 11}

(x)(A âˆª D) âˆ© (B âˆª C) = {3, 5, 7, 9, 11, 15, 17) âˆ© {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 âˆ© B**

**(ii) A âˆ© C**

**(iii) A âˆ© D**

**(iv) B âˆ© C**

**(v) B âˆ© D**

**(vi) C âˆ© D**

**Solution:**

It can be written as

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

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

C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 â€¦}

D = {x: x is a prime number} = {2, 3, 5, 7 â€¦}

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

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

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

(iv)B âˆ© C = Î¦

(v)B âˆ© D = {2}

(vi)C âˆ© 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** â‰¤ x â‰¤ **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 â‰¤ x â‰¤ 6} = {4, 5, 6}

So, we get

{1, 2, 3, 4} âˆ© {4, 5, 6} = {4}

Hence, this pair of sets is not disjoint.

(ii){a, e, i, o, u} âˆ© (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} âˆ© {x: x is an odd integer} = Î¦Hence, 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 â€“ B**

**(ii) A â€“ C**

**(iii) A â€“ D**

**(iv) B â€“ A**

**(v) C â€“ A**

**(vi) D â€“ A**

**(vii) B â€“ C**

**(viii) B â€“ D**

**(ix) C â€“ B**

**(x) D â€“ B**

**(xi) C â€“ D**

**(xii) D â€“ C**

**Solution:**

(i)A â€“ B = {3, 6, 9, 15, 18, 21}

(ii)A â€“ C = {3, 9, 15, 18, 21}

(iii)A â€“ D = {3, 6, 9, 12, 18, 21}

(iv)B â€“ A = {4, 8, 16, 20}

(v)C â€“ A = {2, 4, 8, 10, 14, 16}

(vi)D â€“ A = {5, 10, 20}

(vii)B â€“ C = {20}

(viii)B â€“ D = {4, 8, 12, 16}

(ix)C â€“ B = {2, 6, 10, 14}

(x)D â€“ B = {5, 10, 15}

(xi)C â€“ D = {2, 4, 6, 8, 12, 14, 16}

(xii)D â€“ C = {5, 15, 20}

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

**(i) X â€“ Y**

**(ii) Y â€“ X**

**(iii) X âˆ© Y**

**Solution:**

(i)X â€“ Y = {a, c}

(ii)Y â€“ X = {f, g}

(iii)X âˆ© 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 â€“ Q?**

**Solution:**

We know that

R â€“ Set of real numbers

Q â€“ Set of rational numbers

Hence, R â€“ 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 âˆˆ {2, 3, 4, 5}, 3 âˆˆ {3, 6}

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

(ii)FalseIf a âˆˆ {a, e, i, o, u}, a âˆˆ {a, b, c, d}

So, we get {a, e, i, o, u} âˆ© {a, b, c, d} = {a}

(iii)TrueHere {2, 6, 10, 14} âˆ© {3, 7, 11, 15} = Î¦

(iv)TrueHere {2, 6, 10} âˆ© {3, 7, 11} = Î¦

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