Question 26.31: Simplify the expression : X = AB + A (B + C) + B (B + C)

X = AB + A (B + C) + B (B + C)

Step 1. Applying distributive law to the second and third terms, we have,

X = AB + AB + AC + BB + BC

Step 2.       Now BB = B and AB + AB = AB so that :

X = AB + AC + B + BC

Step 3.        B + BC = B(1 + C) = B⋅1 = B

∴       X = AB + AC + B

Step 4. Factoring B out, we have,

X = B(A + 1) + AC

Step 5.   A + 1 = 1 so that B( A + 1) = B⋅1 = B.

∴       X = B + AC

The original expression is simplified as far as it can go. Once you get acquainted with Boolean simplification techniques, you can combine many individual steps.