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**1. Find the union of each of the following pairs of sets:** **(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} B = {x : x is a natural number less than 6}**

Upon simplifying sets A and B can be rewritten as

A = {3, 6, 9, 12, 15, ………}

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

A ∪ B = {1, 2, 3, 4, 5, 6, 9, 12, 15, ………..}**(iv) A = {x : x is a natural number and 1 < x ≤6 } B = {x : x is a natural number and 6 < x < 10 } **

Upon simplifying sets A and B can be rewritten as

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

B = {7, 8, 9}

A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}**(v) A = {1, 2, 3}, B = φ**

A ∪ B = {1, 2, 3} ∪ φ = {1, 2, 3}

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

Given

A = {a, b}

B = {a, b, c}

Yes A ⊂ B as every element in set A is also in Set B

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

A ∪ B = {a, b, c}

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

Because A is subset of B

Every element of A is an element of B

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 **

A ∪ B = {1, 2, 3, 4} ∪ {3, 4, 5, 6} = {1, 2, 3, 4, 5, 6}

A ∪ B = {1, 2, 3, 4, 5, 6}

**(ii) A ∪ C**

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

A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

**(iii) B ∪ C**

B ∪ C = {3, 4, 5, 6} ∪ {5, 6, 7, 8} = {3, 4, 5, 6, 7, 8}

B ∪ C = {3, 4, 5, 6, 7, 8}

**(iv) B ∪ D**

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

B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

**(v) A ∪ B ∪ C**

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

A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

**(vi) A ∪ B ∪ D**

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}

A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

**(vii) B ∪ C ∪ D**

B ∪ C ∪ D = {3, 4, 5, 6} ∪ {5, 6, 7, 8} ∪ {7, 8, 9, 10} = {3, 4, 5, 6, 7, 8, 9, 10}

B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

**5. 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 **

A ∩ B = {1, 2, 3, 4} ∩ {3, 4, 5, 6} = {3, 4}

A ∩ B = {3, 4}

**(ii) A ∩ C**

A

**∩**C = {1, 2, 3, 4}

**∩**{5, 6, 7, 8} = φ

A

**∩**C = φ

**(iii) B ∩ C**

B

**∩**C = {3, 4, 5, 6}

**∩**{5, 6, 7, 8} = {5, 6}

B

**∩**C = {5, 6}

**(iv) B ∩ D**

B

**∩**D = {3, 4, 5, 6}

**∩**{7, 8, 9, 10} = φ

B

**∩**D = φ

**(v) A ∩ B ∩ C**

A

**∩**B

**∩**C = {1, 2, 3, 4}

**∩**{3, 4, 5, 6}

**∩**{5, 6, 7, 8} = φ

A

**∩**B

**∩**C = φ

**(vi) A ∩ B ∩ D**

A

**∩**B

**∩**D = {1, 2, 3, 4}

**∩**{3, 4, 5, 6}

**∩**{7, 8, 9, 10} = φ

A

**∩**B

**∩**D = φ

**(vii) B ∩ C ∩ D**

B

**∩**C

**∩**D = {3, 4, 5, 6}

**∩**{5, 6, 7, 8}

**∩**{7, 8, 9, 10} = φ

B

**∩**C

**∩**D = 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 **

A ∩ B = {3, 5, 7, 9, 11} ∩ {7, 9, 11, 13} = {7, 9, 11}

A ∩ B = {7, 9, 11}**(ii) B ∩ C **

B ∩ C = {7, 9, 11, 13} ∩ {11, 13, 15} = {11, 13}

B ∩ C = {11, 13}**(iii) A ∩ C ∩ D**

A ∩ C ∩ D = {3, 5, 7, 9, 11} ∩ {11, 13, 15} ∩ {15, 17} = φ

A ∩ C ∩ D = φ**(iv) A ∩ C**

A ∩ C = {3, 5, 7, 9, 11} ∩ {11, 13, 15} = {11}

A ∩ C = {11}**(v) B ∩ D**

B ∩ D = {7, 9, 11, 13} ∩ {15, 17} = φ

B ∩ D = φ

(v**i) A ∩ (B ∪ C)**

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} = {7, 9, 11}

A ∩ (B ∪ C) = {7, 9, 11}**(vii) A ∩ D**

A ∩ D = {3, 5, 7, 9, 11} ∩ {15, 17} = φ

A ∩ D = φ**(viii) A ∩ (B ∪ D) **

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}

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

(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}

(A ∩ B) ∩ (B ∪ C) = {7, 9, 11}

(**x) (A ∪ D) ∩ ( B ∪ C)**

(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}

(A ∪ D) ∩ (B ∪ C) = {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}D = {x : x is a prime number}Then Find**

**(i) A ∩ B**

A ∩ B = {x : x is a natural number} ∩ {x : x is an even natural number}

A ∩ B = {x : x is an even natural number}

**(ii) A ∩ C**

A ∩ C = {x : x is a natural number} ∩ {x : x is an odd natural number}

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

**(iii) A ∩ D**

A ∩ D = {x : x is a natural number} ∩ {x : x is a prime number}

A ∩ D = {x : x is a prime number}

**(iv) B ∩ C**

B ∩ C = {x : x is an even natural number} ∩ {x : x is an odd natural number}

B ∩ C = φ

**(v) B ∩ D**

B ∩ D = {x : x is an even natural number} ∩ {x : x is a prime number}

B ∩ D = {2}

**(vi) C ∩ D**

C ∩ D = {x : x is an odd natural number} ∩ {x : x is a prime number}

C ∩ D = {2}

**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}**

Let’s first simplify {x : x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

So {1, 2, 3, 4} and {4, 5, 6} have nothing in common hence these are disjoint**(ii) {a, e, i, o, u} and {c, d, e, f}**

Sets {a, e, i, o, u} and {c, d, e, f} have element *e* in common hence these two sets are not disjoint**(iii) {x : x is an even integer} and {x : x is an odd integer}**

Sets {x : x is an even integer} and {x : x is an odd integer} does not have anything in common that’s why these two disjoint sets.

**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**

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

A – B = {3, 6, 9, 15, 18, 21}

**(ii) A – C**

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

A – C = {3, 9, 15, 18, 21}

**(iii) A – D**

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

A – D = {3, 6, 9, 12, 18, 21}

**(iv) B – A**

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

B – A = {4, 8, 16, 20}

**(v) C – A**

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

C – A = {2, 4, 8, 10, 14, 16}

**(vi) D – A**

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

D – A = {5, 10, 20}

**(vii) B – C**

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

B – C = {20}

**(viii) B – D**

B – D = {4, 8, 12, 16, 20} – {5, 10, 15, 20}

B – D = {4, 8, 12, 16}

**(ix) C – B**

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

C – B = {2, 6, 10, 14}

**(x) D – B**

D – B = {5, 10, 15, 20} – {4, 8, 12, 16, 20}

D – B = {5, 10, 15}

**(xi) C – D**

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

C – D = {2, 4, 6, 8, 12, 14, 16}

**(xii) D – C**

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

D – C = {5, 15, 20}

**10. If X= {a, b, c, d} and Y = {f, b, d, g} Find(i) X – Y**

X – Y = {a, b, c, d} – {f, b, d, g} = {a, c}

X – Y = {a, c}

**(ii) Y – X**

Y – X = {f, b, d, g} – {a, b, c, d} = {f, g}

Y – X = {f, g}

**(iii) X ∩ Y**

X ∩ Y = {a, b, c, d} ∩ {f, b, d, g} = {b, d}

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?**

R = Set of Real Numbers = Set of Rational and Irrational Numbers

Q = Set of Rational Numbers

R – Q = Set of Rational and Irrational Numbers – Set of Rational Numbers*R – Q = 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**

False**(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets**

False**(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets**

False**(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets**

False