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 Є 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) 𝜙’ ∩ A = ∩ A = A

(iii) A ∩ A’ = φ™

(iv) ∪’ ∩ A = φ™ ∩ A = φ™