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

3.4 BOOLEAN AXIOMS AND THEOREMS

The basic logic operations include logic sum, logic product, and logic complement. If a logic variable is true, its logic complement is false. The following set of logic expressions illustrates the axioms of Boolean algebra:

• 0 * 0 = 0
• 0 + 0 = 0
• 1 * 1 = 1
• 1 + 1 = 1
• 0 * 1 = 1 * 0 = 0
• 0 + 1 = 1 + 0 = 1
• if x = 0, then = 1
• if x = 1; then = 0

The character (*) represents the AND logic product, and the character (+) stands for the OR logic sum. A bar over a character represents the NOT logic. From these logic axioms, basic Boolean identities were formulated. The following expressions illustrate these identities.

Identity Property

• x + 0 = x
• x * 1 = x
• x + 1 = 1
• x * 0 = 0

Idempotent Property

• x + x = x
• x * x = x

Complement Property

• x + = 1
• x * = 0

Involution Property

Commutative Property

• x + y = y + x
• x * y = y * x

Associative Property

• x + (y + z) = (x + y) + z
• x * (y * z) = (x * y) * z

Distributive Property

• x * (y + z) = (x * y) + (x * z)
• x + (y * ...