State and prove following laws of Boolean algebra.(a) De-Morgan’s Law(b) Distributive Law

(a) De-Morgan’s Law:

De-Morgan's law is an important rule in Boolean algebra and it consists of two laws.

1. The complement of the sum of two variables is equal to the product of their complements. Mathematically, it can be represented as:

   (A + B)' = A' . B'

2. The complement of the product of two variables is equal to the sum of their complements. Mathematically, it can be represented as:

   (A . B)' = A' + B'

Proof:

Let's prove the first law. We know that A + A' = 1 (complement law). Now, if we multiply both sides by B', we get:

B'(A + A') = B' * 1 B'A + B'A' = B'

Now, if we add A'B' to both sides, we get:

B'A + A'B' + B'A' = B' + A'B'

By rearranging, we get:

B'A + A'B' = (A + B)'

This proves the first law.

The proof of the second law is similar and left as an exercise.

(b) Distributive Law:

The distributive law states that the product of a variable and the sum of two other variables is equal to the sum of the product of the variable and each of the other variables. It can be represented as:

A . (B + C) = (A . B) + (A . C)

Proof:

Let's prove this law. We know that A . 1 = A and A . 0 = 0 (identity laws). Now, if we replace 1 with (B + B') and 0 with (C + C'), we get:

A . (B + C) = A . [(B + B') + (C + C')]

By rearranging, we get:

A . (B + C) = (A . B) + (A . C)

This proves the distributive law.