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**Exercise 1.5**

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

**(i) A’ **

**(ii) B’ **

**(iii) (A C)’ **

**(iv) (A B)’**

**(v) (A’)’ **

**(vi) (B – C)’**

**Ans.** Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},

A = {1, 2, 3, 4},

B = {2, 4, 6, 8} and C = {3, 4, 5, 6}.

**(i) **A’ = U – A = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {1, 2, 3, 4}

= {5, 6, 7, 8, 9}

**(ii) **B’ = U – B = {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2, 4, 6, 8}

= {1, 3, 5, 7, 9}

**(iii) **(A C)’ = U – (A C)

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

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

**(iv) **(A B)’ = U – (A B)

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

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

**(v) **(A’)’ = U – A’ = U – (U – A)

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

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

= {1, 2, 3, 4} = A

**(vi) **(B – C)’ = U – (B – C)

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

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

= {1, 3, 4, 5, 6, 7, 9}

**2. If U = **** find the complement of the following sets:**

**(i) A = **

**(ii) B = **

**(iii) C = **

**(iv) D = **

**Ans. **Given: U =

**(i) **A’ = U – A

=

**(ii) **B’ = U – B

=

**(iii) **C’ = U – C

=

**(iv) **D’ = U – D

=

**3. Taking the set of natural numbers as the universal set, write down the complement of the following set:**

**(i) {**** is an even natural number}**

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

**(iii) {**** is a positive multiple of 3}**

**(iv) {**** is a prime number}**

**(v) {**** is a natural number divisible by 3 and 5}**

**(vi) {**** is a perfect square}**

**(vii) {**** is a perfect cube}**

**(viii) {**** + 5 = 8}**

**(ix) {**** +5=9}**

**(x) **

**(xi) {** ** N and ****}**

**Ans. **Given: U =

**(i) **Let A = { is an even natural number}

A’ = U – A = – { is an even natural number}

= { is an odd natural number}

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

A’ = U – A = – { is an odd natural number}

= { is an even natural number}

**(iii) **Let A = { is a positive multiple of 3}

A’ = U – A = – { is a positive multiple of 3}

= { is not a positive multiple of 3}

**(iv) **Let A = { is a prime number}

A’ = U – A = – { is a prime number}

= { is not a prime number}

**(v) **Let A = { is a natural number divisible by 3 and 5}

A’ = U – A = – { is a natural number divisible by 15}

= { is not divisible by 15}

**(vi) **Let A = { is a perfect square}

A’ = U – A = – { is a perfect square}

= { is not a perfect square}

**(vii) **Let A = { is a perfect cube}

A’ = U – A = – { is a perfect cube}

= { is not a perfect cube}

**(viii) **Let A = { + 5 = 8} = {3}

A’ = U – A = – {3}

= { 3}

**(ix) **Let A = { +5=9} = {2}

A’ = U – A = – {2}

= { 2}

**(x) **Let A = = {7, 8, 9, 10, ………}

A’ = U – A = – {7, 8, 9, 10, ………}

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

**(xi) **Let A = { N and } = {5, 6, 7, 8, ………..}

A’ = U – A = – {5, 6, 7, 8, ………..}

= {1, 2, 3, 4}

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

**(i) **

**(ii) **

**Ans. **Given: U = {1, 2, 3, 4, 5, 6, 7, 8, 9},

A = {2, 4, 6, 8} and B = {2, 3, 5, 7}

**(i) **L.H.S. = = U – (A B)

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

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

R.H.S. = = (U – A) (U – B)

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

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

L.H.S. = R. H. S.

**(ii) **L.H.S. =

= U – (A B)

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

= {1, 2, 3, 4, 5, 6, 7, 8, 9} – {2}

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

R.H.S. = = (U – A) (U – B)

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

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

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

L.H.S. = R. H. S.

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

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i) **In the diagrams, shaded portion represents

**(ii) **In the diagrams, shaded portion represents

**(iii) **In the diagrams, shaded portion represents

**(iv) **In the diagrams, shaded portion represents

**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 **** what is A’?**

**Ans. **Given: U = { is a triangle}

A = { is a triangle and has at least one angle different from }

A’ = U – A = { is a triangle and has all angles equal to }

= Set of all equilateral triangles

**7. Fill in the blanks to make each of the following a true statement:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**Ans. (i)**

**(ii)**

**(iii)**

**(iv)**

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