# NCERT Solutions Excercise 5

**Exercise- 1.5**

**1. Let = {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’ = **{5, 6, 7, 8, 9}.

**(ii) B’ = **{1, 3, 5, 7, 9}.

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

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

**(v) (A’)’ = **{1, 2, 3, 4}

**(vi) (B – C)’ = **{2, 4, 6, 8} – {3, 4, 5, 6}

(B – C) = {2, 8}

(B – C)’ = {1, 3, 4, 5, 6, 7, 9}.

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

**A = {a, b, c} = **A’ = {d, e, f, g, h}

**B = {d, e, f, g} =** B’ = {a, b, c, h}

**C = {a, c, e, g} =** C’ = {b, d, f, h}

**D = {f, g, h, a} = **D’ = {b, c, d, e}

**3. Taking the set of natural numbers as universal set, write drown the compliments of the following sets:**

**(i) {x : x is an even number}.**

= {1, 2, 3, 4, 5, 6, …..}

Let A = {x : x is an even natural no.}

= {2, 4, 6, 8, 10, 12}

A’ = {x : x is an odd natural no.}

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

= {1, 2, 3, 4, 5, 6, …..}

Let A = {x : x is an odd natural no.}

= {1, 3, 5, 7, 9, ……}.

A’ = {x : x is an even natural no.}.

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

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

Let A = {x : x is a multiple of 3}

= {3, 6, 9, 12, 15, 18, ……}.

A’ = {x : x is Ð„ N and x is not a multiple of 3}.

**(iv) {x : x is a prime no}.**

= {1, 2, 3, 4, 5, 6, …..}

Let A = {x : x is a prime number}.

= {2, 3, 5, 7, 11, ……}.

A’ = {x : x is a positive composite number and x = 1}.

**(v) {x : x is a natural no. divisible by 3 and 5}.**

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

Let A = {x : x is a natural no. divisible by 3 and 5}.

= {3, 5, 6, 9, 10, 12, ……}.

A’ = {x : x is a positive integer which is not divisible by 3 or not divisible by 5}.

**(vi) {x : x is a perfect square}.**

= {1, 2, 3, 4, 5, 6, …..}

Let A = {x : x is a perfect square}.

= {2, 9, 16, 25, ……}.

A’ = {x : x N and x is not a perfect square}.

**(vii) {x : x is a perfect cube}.**

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

Let A = {x : x is a perfect cube}.

= {8, 27, …..}.

A’ = {x : x N and x is not a perfect cube}.

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

= {1, 2, 3, 4, 5, 6, ……}.

Let A = {x : x + 5 = 8}.

= x = 3.

A’ = {x : x N and x ≠ 3}.

**(ix) {x : 2x + 5 = 9}**

= {1, 2, 3, 4, 5, 6, ……}.

Let A = {x : 2x + 5 = 9}.

x = 2

A’ = {x : x N and x ≠ 2}.

**(x) {x : x ****≥ 7}**

= {1, 2, 3, 4, 5, 6, ……}.

Let A = {x : x ≥ 7}.

= {7, 8, 9, 10, ……}.

A’ = {x : x N and x < 7}.

**(xi) {x : x N and 2x + 1 > 10}**

= {1, 2, 3, 4, 5, 6, ……}.

Let A = {x : x N and 2x + 1 > 10}.

= 2x + 1 > 10

2x > 10 – 1

x=9/2

**4. If **** = {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’**

Taking L.H.S Taking R.H.S

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

(A ∪ B)’ = {1, 9} B’ = {1, 4, 6, 8, 9}

∴ A ∩ B = {1, 9}

Hence Proved

**(ii) (A ∩ B)’**** = A’** **∪** **B’**

Taking L.H.S Taking R.H.S

A ∩ B = {2} A’ = {1, 3, 5, 7, 9}

(A ∩ B)’ = {1, 3, 4, 5, 6, 7, 8, 9} B’ = {1, 4, 6, 8, 9} ∴ A’ ∪ B’ = {1, 3, 4, 5, 6, 7, 8, 9}.

Hence Proved

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

**(i) (A** **∪** **B)’**

**(ii) A’ ∩ B’**

**(iii) (A ∩ B)’**

**(iv) ****A’** **∪** **B’**

formulas :- (i) (A ∪ B)’ = A’ ∩ B’

(ii) (A ∩ B)’ = A’ ∪ B’

Ans :- Shaded part

**6. Let ∪ 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’?**

A’ = The set of all the triangles which has different angle from 60.

= The set of all the triangles has each angle is 60.

∴ The set of all the equilateral triangle.

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

**(i) ****A ****∪ A’ = **

**(****ii)** **ðÂœ™’ ∩ A = ** ∩ A = A

**(****iii) A**** ∩ A’ = **φ™

**(iv) ****∪****’ ∩**** A = **φ™ ∩ A = φ™