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**Exercise 1.4**

**1. Find the union of each of the following pairs of sets:**

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

**(ii) A = **** and B = **

**(iii) A = {**** is a natural number and multiple of 3} and B = {**** is a natural number less than 6}**

**(iv) A = {**** is a natural number and ****} and B = {**** is a natural number and ****}**

**(v) A = {1, 2, 3} and B = **

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

**(ii) **A B =

**(iii) **A B = {1, 2, 3, 4, 5, 6, 9, 12, 15, ………..}

**(iv) **A B = {2, 3, 4, 5, 6, 7, 8, 9}

**(v) **A B = {1, 2, 3}

**2. Let A = **** and B = ****. Is A **** B? What is A **** B?**

**Ans. **Given: A = and B = .

Here all elements of set A are present in set B.

A B and A B = = B

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

**Ans. **Given: A and B are two sets such that A B

Taking A = {1, 2} and B = {1, 2, 3}, then A B = {1, 2, 3} = 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**

**Ans. **Given: 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} {3, 4, 5, 6}

= {1, 2, 3, 4, 5, 6}

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

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

**(iii) **B C = {3, 4, 5, 6} {5, 6, 7, 8}

= {3, 4, 5, 6, 7, 8}

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

= {3, 4, 5, 6, 7, 8, 9, 10}

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

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

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

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**(vii) **B C D = {3, 4, 5, 6} {5, 6, 7, 8} {7, 8, 9, 10}

= {3, 4, 5, 6, 7, 8, 9, 10}

**5. Find the intersections of each pair of sets of question 1 above.**

**Ans. (i) **X Y = {1, 3}

**(ii) **A B =

**(iii) **A B = {3, 6, 9, 12, ………..}

**(iv) **A B =

**(v) **A B =

**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)**

**Ans. **Given: A = {3, 5, 7, 9, 11},

B = {7, 9, 11, 13},

C = {11, 13, 15} and D = {15, 17}

**(i) **A B = {3, 5, 7, 9, 11} {7, 9, 11, 13}

= {7, 9, 11}

**(ii) **B C = {7, 9, 11, 13} {11, 13, 15}

= {11, 13}

**(iii) **A C D

= {3, 5, 7, 9, 11} {11, 13, 15} {15, 17} =

**(iv) **A C = {3, 5, 7, 9, 11} {11, 13, 15}

= {11}

**(v) **B D = {7, 9, 11, 13} {15, 17} =

**(vi) **A (B C)

= {3, 5, 7, 9, 11} ({7, 9, 11, 13} {11, 13, 15})

= {3, 5, 7, 9, 11} {7, 9, 11, 13, 15, 17} = {7, 9, 11}

**(vii) **A C = {3, 5, 7, 9, 11} {15, 17} =

**(viii) **A (B D)

= {3, 5, 7, 9, 11} ({7, 9, 11, 13} {15, 17})

= {3, 5, 7, 9, 11} {7, 9, 11, 13, 15, 17}

= {7, 9, 11}

**(ix) **(A B) (B C)

= ({3, 5, 7, 9, 11} {7, 9, 11, 13}) ({7, 9, 11, 13} {11, 13, 15})

= {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} {11, 13, 15})

= {3, 5, 7, 9, 11, 15, 17} {7, 9, 11, 13, 15}

= {7, 9, 11, 15}

**7. If A = {**** is a natural number}, B = {**** is an even natural number}, C = {**** is an odd natural number} and D = {**** is a prime number}, find:**

**(i) A B **

**(ii) A C **

**(iii) A D **

**(iv) B C **

**(v) B D **

**(vi) C D **

**Ans. (i) **A B = { is a natural number} { is an even natural number} = B

**(ii) **A C = { is a natural number} { is an odd natural number} = C

**(iii) **A D = { is a natural number} { is a prime number} = D

**(iv) **B C = { is an even natural number} { is an odd natural number} =

**(v) **B D = B C = { is an even natural number} { is a prime number} = {2}

**(vi) **C D = { is an odd natural number} { is a prime number}

= { is an odd prime number}

**8. Which of the following pair of sets are disjoint:**

**(i) {1, 2, 3, 4} and {**** is a natural number and ****}**

**(ii) **** and **

**(iii) {**** is an even integer} and {**** is an odd integer}**

**Ans. (i) **Let A = {1, 2, 3, 4} and B = { is a natural number and } = {4, 5, 6}

A B = {4}

Therefore, A and B are not disjoint.

**(ii) **Let A = and B =

A B =

Therefore, A and B are disjoint.

**(iii) **Let A = { is an even integer} and B = { is an odd integer}

A B =

Therefore, A and B are 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 **

**Ans. **Given: 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};

**(i) **A – B = {3, 6, 9, 12, 15, 18, 21} – {4, 8, 12, 16, 20}

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

**(ii) **A – B = {3, 6, 9, 12, 15, 18, 21} – {2, 4, 6, 8, 10, 12 , 14, 16}

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

**(iii) **A – B = {3, 6, 9, 12, 15, 18, 21} – {5, 10, 15, 20}

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

**(iv) **B – A = {4, 8, 12, 16, 20} – {3, 6, 9, 12, 15, 18, 21}

= {4, 8, 16, 20}

**(v) **C – A = {2, 4, 6, 8, 10, 12 , 14, 16} – {3, 6, 9, 12, 15, 18, 21}

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

**(vi) **D – A = {5, 10, 15, 20} – {3, 6, 9, 12, 15, 18, 21}

= {5, 10, 20}

**(vii) **B – C = {4, 8, 12, 16, 20} – {2, 4, 6, 8, 10, 12 , 14, 16} = {20}

**(viii) **B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20}

= {4, 8, 12, 16}

**(ix) **C – B = {2, 4, 6, 8, 10, 12 , 14, 16} – {4, 8, 12, 16, 20}

= {2, 6, 10, 14}

**(x) **D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20}

= {5, 10, 15}

**(xi) **C – D = {2, 4, 6, 8, 10, 12 , 14, 16} – {5, 10, 15, 20}

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

**(xii) **D – C = {5, 10, 15, 20} – {2, 4, 6, 8, 10, 12 , 14, 16}

= {5, 15, 20}

**10. If X = **** and Y = **** find:**

**(i) X – Y **

**(ii) Y – X **

**(iii) X **** Y**

**Ans. **Given: X = and Y =

**(i) **X – Y = =

**(ii) **Y – X = =

**(iii) **X Y = =

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

**Ans. **We know that set of real numbers contain rational and irrational numbers.

Therefore, R – Q = set of irrational numbers.

**12. State whether each of the following statements is true or false. Justify your answer.**

**(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.**

**(ii) **** and **** 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.**

**Ans. (i) **Let A = {2, 3, 4, 5} and B = {3, 6}

A B = {3}

A and B are not disjoint. Therefore, statement is false.

**(ii) **Let A = and B =

A B =

A and B are not disjoint. Therefore, statement is false.

**(iii) **Let A = {2, 6, 10, 14} and B = {3, 7, 11, 15}

A B =

A and B are disjoint. Therefore, statement is true.

**(iv) **Let A = {2, 6, 10} and B = {3, 7, 11}

A B =

A and B are disjoint. Therefore, statement is true.

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