Simplification using Boolean algebra

Let us consider an example of a Boolean function: AB+A (B+C) + B (B+C)

The logic diagram for the Boolean function AB+A (B+C) + B (B+C) can be represented as:

We will simplify this Boolean function on the basis of rules given by Boolean algebra.

AB + A (B+C) + B (B+C)

AB + AB + AC + BB + BC     {Distributive law; A (B+C) = AB+AC, B (B+C) = BB+BC}

AB + AB + AC + B + BC         {Idempotent law; BB = B}

AB + AC + B + BC             {Idempotent law; AB+AB = AB}

AB + AC +B             {Absorption law; B+BC = B}

B + AC             {Absorption law; AB+B = B}

Hence, the simplified Boolean function will be B + AC.

The logic diagram for Boolean function B + AC can be represented as: