Question

If= {1,2,3,4,5,6,7,8,9},A = {2,3,5,7,9},B = { 1,2,4,6}, verify:
a. (A U B)' = A' ∩ B'
b. B – A = B ∩ A' = B – (A ∩ B)​

Answer 1

Given:

• Union set = U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
• Set A = {2, 3, 5, 7, 9}
• Set B = {1, 2, 4, 6}

To prove:

• (A U B)' = A' ∩ B'
• B – A = B ∩ A' = B – (A ∩ B)

Solution:

There are 2 parts in this question.

Proof for 1st part:

First, consider the L.H.S

(A U B)' = Compliment All the numbers in Set A and Set B

⇒ (A U B)' = {1, 2, 3, 4, 5, 6, 7, 9}'

LHS → (A U B)' = 8

Now consider R.H.S part.

• A' = {1, 4, 6, 8}
• B' = {3, 5, 7, 8, 9}

A' ∩ B'

RHS = 8

LHS = RHS

Hence (a) proved

____________________________

Proof for 2nd part:

B - A = {1, 4, 6} ----- [Equation 1]

B ∩ A' = {1, 2, 4, 6} ∩ {1, 4, 6, 8}

→ B ∩ A' = {1, 4, 6} ----- [Equation 2]

B - (A ∩ B)

= B - {2}

= {1, 2, 4, 6} - {2}

= {1, 4, 6} ----- [Equation 3]

Equation 1 = Equation 2 = Equation 3

★ Hence (b) proved ★