State and prove following laws of Boolean algebra.(a) Consensus Law(b) Absorption Law

(a) Consensus Law:

The Consensus Law in Boolean algebra states that for three variables A, B, and C:

A + BC = (A + B)(A + C)

Proof:

We can prove this law by using the basic laws of Boolean algebra.

Step 1: Start with the right-hand side of the equation:

(A + B)(A + C)

Step 2: Apply the Distributive Law:

= AA + AC + BA + BC

Step 3: Apply the Idempotent Law (AA = A) and the Commutative Law (BA = AB):

= A + AC + AB + BC

Step 4: Apply the Distributive Law again:

= A(1 + C + B) + BC

Step 5: Apply the Null Law (A + A' = 1) and the Identity Law (A1 = A):

= A + BC

This proves the Consensus Law.

(b) Absorption Law:

The Absorption Law in Boolean algebra states that for two variables A and B:

A + AB = A A(A + B) = A

Proof:

We can prove this law by using the basic laws of Boolean algebra.

For the first equation:

Step 1: Start with the left-hand side of the equation:

A + AB

Step 2: Apply the Distributive Law:

= A(1 + B)

Step 3: Apply the Null Law (A + A' = 1):

= A

This proves the first equation of the Absorption Law.

For the second equation:

Step 1: Start with the left-hand side of the equation:

A(A + B)

Step 2: Apply the Distributive Law:

= AA + AB

Step 3: Apply the Idempotent Law (AA = A):

= A + AB

Step 4: Apply the Absorption Law (from the first equation):

= A

This proves the second equation of the Absorption Law.