Verify De Morgan’s laws for set difference using the sets given below:

A = {1, 3, 5, 7, 9, 11, 13, 15}, B = {1, 2, 5, 7} and C = {3, 9, 10, 12, 13}.

We know that De Morgan’s laws for set difference are:

(a) A (B ⋃ C) = (A B) ⋂ (A C)

(b) A (B ⋂ C) = (A B) ⋃ (A C)

Case (a):

First, we shall verify (a). To do this consider,

B ⋃ C = {1, 2, 5, 7} ⋃ {3, 9, 10, 12, 13}

= {1, 2, 3, 5, 7, 9, 10, 12, 13}

⇒ A (B ⋃ C) = A {1, 2, 3, 5, 7, 9, 10, 12, 13}

= {11, 15} … (1)

Now, A B = {1, 3, 5, 7, 9, 11, 13, 15} {1, 2, 5, 7}

= {3, 9, 11, 13, 15}

And A C = {1, 3, 5, 7, 9, 11, 13, 15} {3, 9, 10, 12, 13}

= {1, 5, 7, 11, 15}

Then, (A B) ⋂ (A C) = {3, 9, 11, 13, 15} ⋂ {1, 5, 7, 11, 15}

= {11, 15} … (2)

From (1) and (2), it follows that A (B ⋃ C) = (A B) ⋂ (A C).

Case (b):

Next, we shall verify (b). To do this consider,

B ⋂ C = {1, 2, 5, 7} ⋂ {3, 9, 10, 12, 13}

= ϕ

⇒ A (B ⋂ C) = A ϕ

= {1, 3, 5, 7, 9, 11, 13, 15} … (3)

Now, A B = {1, 3, 5, 7, 9, 11, 13, 15} {1, 2, 5, 7}

= {3, 9, 11, 13, 15}

And A C = {1, 3, 5, 7, 9, 11, 13, 15} {3, 9, 10, 12, 13}

= {1, 5, 7, 11, 15}

Then, (A B) ⋃ (A C) = {3, 9, 11, 13, 15} ⋂ {1, 5, 7, 11, 15}

= {1, 3, 5, 7, 9, 11, 13, 15} … (4)

From (3) and (4), it follows that A (B ⋂ C) = (A B) ⋃ (A C).

Hence, De Morgan’s laws of set difference are verified.