Computers are built based on digital logic circuits. Here the word logic indicates the role of logic in the design of circuits. The word digital indicates that the circuits process discrete, or separate, signals as opposed to continuous ones. Electronic engineers describe the true or false condition of electronic switches using binary digits 1 and 0, called bits, instead of T and F.

Simple electronic circuits such as switches in series or in parallel correspond directly to their logical counterparts Ù and Ú . An AND-gate is a circuit with two input signals and one output signal. If both input signals are 1, then the output signal is 1. Otherwise, the output signal is 0. An OR-gate also has two input signals and one output signal. If both input signals are 0, then the output signal is 0. Otherwise, the output signal is 1.

Determining Output for a Given Input
Given a set of input signals for a circuit the output can be determined as in Example 1.4.1 by tracing through the circuit gate by gate.

Constructing an Input/Output Table for a Circuit top
To construct the input/output table for a circuit, first determine the number of rows required based on the number, n, of input variables. We know that all combinations can be written down in 2ⁿ rows. Now find the output for each combination of input variables as in Example 1.4.2.

In honor of the English mathematician George Boole, any variable, such as a statement variable or an input signal, that can take one of only two values is called a Boolean variable. An expression composed of Boolean variables and the connectives Ø , Ù , and Ú is called a Boolean expression.

Finding a Boolean Expression for a Circuit top
To find the Boolean expression for a circuit, trace through the circuit from left to right, indicating the output of each gate symbolically. See Example 1.4.3.b. The symbolic representation for this circuit is (P Ù Q) Ù Ø R, commonly called a recognizer circuit. A recognizer is a circuit that outputs a 1 for exactly one combination of input signals and outputs 0's for all other combinations of input signals.

Constructing Circuits for Boolean Expressions top
To construct circuits for Boolean expressions

1. Write the input variables in a column on the left side of the diagram.
2. Working from the right side of the diagram, determine the last operation performed in the expression and place the appropriate gate at the extreme right of the diagram.
3. Continuing to work from the outermost part of the expression to the innermost part, determine appropriate gate(s) for the input to the gate on the right. Unary operators have a single input while binary operators have two inputs.
4. Progressively repeat steps 2 and 3 until the inputs are the variables on the left side of the diagram.