**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** ...

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