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

Involution Property

images

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 * ...

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