Question

Write equivalent boolean expression for the following logic circuit:

Answer 1

Let’s begin with a semiconductor gate circuit in need of simplification. The “A,” “B,” and “C” input signals are assumed to be provided from switches, sensors, or perhaps other gate circuits. Where these signals originate is of no concern in the task of gate reduction.

How to Write a Boolean Expression to Simplify Circuits

Our first step in simplification must be to write a Boolean expression for this circuit. This task is easily performed step by step if we start by writing sub-expressions at the output of each gate, corresponding to the respective input signals for each gate. Remember that OR gates are equivalent to Boolean addition, while AND gates are equivalent to Boolean multiplication. For example, I’ll write sub-expressions at the outputs of the first three gates:

. . . then another sub-expression for the next gate:

Finally, the output (“Q”) is seen to be equal to the expression AB + BC(B + C):

Now that we have a Boolean expression to work with, we need to apply the rules of Boolean algebra to reduce the expression to its simplest form (simplest defined as requiring the fewest gates to implement):

The final expression, B(A + C), is much simpler than the original, yet performs the same function. If you would like to verify this, you may generate a truth table for both expressions and determine Q’s status (the circuits’ output) for all eight logic-state combinations of A, B, and C, for both circuits. The two truth tables should be identical.