Theory:

A simplified Boolean expression uses the fewest gates possible to implement a given expression.

Example Using Boolean algebra techniques, simplify this expression:

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

Solution:

1. Step 1: Apply the distributive law to the second and third terms in the expression, as follows:

$$A B + A B + A C + B B + B C$$

2. Step 2: Apply rule 7 (BB = B) to the fourth term.

$$AB + AB + AC + B + BC$$

3. Step 3: Apply rule 5 (AB + AB = AB) to the first two terms.

$$AB + AC + B + BC$$

4. Step 4: Apply rule 10 (B + BC = B) to the last two terms.

$$AB + AC + B$$

5. Step 5: Apply rule 10 (AB + B = B) to the first and third terms.

$$B+AC$$

At this point the expression is simplified as much as possible.