Introduction to Digital Systems: Modeling, Synthesis, and Simulation Using VHDL

3.2 BOOLEAN THEORY

Boolean theory provides the basic fundamentals for logic operators and operations to perform Boolean algebra. Boolean algebra is a branch of mathematics that includes methods for manipulating logical variables and logical expressions. The Greek philosopher Aristotle founded a system of logic based on only two types of propositions: true and false. His bivalent (two-mode) definition of truth led to the four foundational laws of logic: the Law of Identity (A is A); the Law of Noncontradiction (A is not non-A); the Law of the Excluded Middle (either A or non-A); and the Law of Rational Inference. These “laws” function within the scope of logic where a proposition is limited to one of two possible values, but may not apply in cases where propositions can hold values other than “true” or “false.”

The English mathematician George Boole (1815–1864) sought to give symbolic form to Aristotle's system of logic—hence the name Boolean algebra. Starting with his investigation of the laws of thought, Boole constructed a “logical algebra.” This investigation into the nature of logic and ultimately of mathematics led subsequent mathematicians and logicians into several new fields of mathematics. Two of these, known as the “algebra of propositions” and the “algebra of classes,” were based principally on Boole's work. The algebra now used in the design of logical circuitry is known as Boolean algebra.

In the mid-twentieth century, Claude Shannon, an electrical engineer and mathematician, ...