Digital Logic

For two binary variables (taking values 0 and 1) there are 16 possible functions. The functions involve only three operations which make up Boolean algebra: AND, OR, and COMPLEMENT. They are symbolically represented as follows:

These operations are like ordinary algebraic operations in that they are commutative, associative, and distributive. There is a group of useful theorems of Boolean algebra which help in developing the logic for a given operation.

Boolean Algebra Theorems

The applications of digital logic involve functions of the AND, OR, and NOT operations. These operations are subject to the following identities:

These theorems can be used in the algebraic simplification of logic circuits which come from a straightforward application of a truth table.

Binary Functions of Two Variables

Digital logic involves combinations of the three types of operations for two variables: AND, OR, and NOT. There are sixteen possible functions:

Single Variable Theorems

Two-Variable Theorems

Besides the important DeMorgan's Theorem, the theorems below have utility in digital circuits. They have no direct counterparts in ordinary algebra.