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**1. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9} A = {1, 2, 3, 4}B = {2, 4, 6, 8}C = {3, 4, 5, 6}Find**

**(i) A′**

A′ = {1, 2, 3, 4}’ = {5, 6, 7, 8, 9}

A’ = {5, 6, 7, 8, 9}

**(ii) B′**

B′ = {2, 4, 6, 8}’ = {1, 3, 5, 7, 9}

B′ = {1, 3, 5, 7, 9}

**(iii) (A ∪ C)′**

(A ∪ C)′ = ({1, 2, 3, 4} ∪ {3, 4, 5, 6})’ = {1, 2, 3, 4, 5, 6}’ = {7, 8, 9}

(A ∪ C)′ = {7, 8, 9}

**(iv) (A ∪ B)′**

(A ∪ B)′ = ({1, 2, 3, 4} ∪ {2, 4, 6, 8})’ = {1, 2, 3, 4, 6, 8}’ = {5, 7, 9}

(A ∪ B)′ = {5, 7, 9}

**(v) (A′)′**

(A′)′ = ({1, 2, 3, 4}’)’ = {5, 6, 7, 8, 9}’ = {1, 2, 3, 4}

(A′)′ = A = {1, 2, 3, 4}

**(vi) (B – C)′**

(B – C)′ = ({2, 4, 6, 8} – {3, 4, 5, 6})’ = {2, 8}’ = {1, 3, 4, 5, 6, 7, 9}

(B – C)′ = {1, 3, 4, 5, 6, 7, 9}`

**2. If U = {a, b, c, d, e, f, g, h}Find the complements of the following sets:**

**(i) A = {a, b, c}**

Complement of A = A′ = {a, b, c}′ = {d, e, f, g, h}

**(ii) B = {d, e, f, g}**

Complement of B = B′ = {d, e, f, g}′ = {a, b, c, h}

**(iii) C = {a, c, e, g}**

Complement of C = C′ = {a, c, e, g}′ = {b, d, f, h}

**(iv) D = {f, g, h, a}**

Complement of D = D′ = {f, g, h, a}′ = {b, c, d, e}

**3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:****(i) {x : x is an even natural number} **

{x : x is an odd natural number}**(ii) {x : x is an odd natural number}**

{x : x is an even natural number}**(iii) {x : x is a positive multiple of 3} **

{x : x is not a positive multiple of 3} **(iv) {x : x is a prime number}**

{x : x is a positive composite number or x = 1}**(v) {x : x is a natural number divisible by 3 and 5}**

{x : x is neither a multiple of 3 nor multiple of 5}**(vi) {x : x is a perfect square} **

{x : x is a Natural Number and its not a perfect square}**(vii) {x : x is a perfect cube}**

{x : x is a Natural Number and its not a perfect cube}**(viii) {x : x + 5 = 8} **

{x : x is a Natural Number and x is not equal to 3}**(ix) {x : 2x + 5 = 9}**

{x : x is a Natural Number and x is not equal to 2}**(x) {x : x ≥ 7}**

{x : x is a Natural Number and it’s less than 7} **(xi) {x : x ∈ N and 2x + 1 > 10}**

{x : x is a Natural Number and it’s greater than 4}

**4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}A = {2, 4, 6, 8}B = {2, 3, 5, 7}Verify that**

**(i) (A ∪ B)′ = A′ ∩ B′**

Let’s solve both sides of this equation and then compare

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

A′ ∩ B′ = {2, 4, 6, 8}′ ∩ {2, 3, 5, 7}′ = {1, 3, 5, 7, 9} ∩ {1, 4, 6, 8, 9} = {1, 9}

Hence (A ∪ B)′ = A′ ∩ B′ = {1, 9}

**(ii) (A ∩ B)′ = A′ ∪ B′**

(A ∩ B)′ = ({2, 4, 6, 8} ∩ {2, 3, 5, 7})′ = {2}′ = {1, 3, 4, 5, 6, 7, 8, 9}

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

Hence (A ∩ B)′ = A′ ∪ B′ = {1, 3, 4, 5, 6, 7, 8, 9}

**5. Draw appropriate Venn diagram for each of the following :**

**(i) (A ∪ B)′**

**(ii) A′ ∩ B′**

**(iii) (A ∩ B)′**

**(iv) A′ ∪ B′**

**6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?**

U = {Set of all triangles in a plane}

A = {Set of all triangles in a plane with at least one angle different from 60°}

A′ = U – A = {Set of all Equilateral Triangles}

**7. Fill in the blanks to make each of the following a true statement:****(i) A ∪ A′ = . . .**

A ∪ A′ = U**(ii) φ′ ∩ A = . . .**

φ′ ∩ A = U ∩ A = A**(iii) A ∩ A′ = . . .**

A ∩ A′ = φ**(iv) U′ ∩ A = . . .**

U′ ∩ A = φ ∩ A = φ