## 7

## Boolean Algebra

Boolean algebra is a significant tool for the analysis and design of electronic computers. It has wide applications to switching theory and logical design of electronic circuits.

##### 7.1 DEFINITIONS AND BASIC PROPERTIES

**Definition 7.1**

A non-empty set *B* with two binary operations ∨ and ∧, a unary operation ′, and two distinct elements 0 and *I* is called a **Boolean Algebra** if the following axioms holds for any elements *a*, *b*, *c* ∈ *B*:

[*B*_{1}]: **Commutative Laws**

*a* ∨ *b* = *b* ∨ *a* and *a* ∧ *b* = *b* ∧ *a*.

[*B*_{2}]: **Distributive Laws**

*a* ∧ (*b* ∨ *c*) = (*a* ∧ *b*) ∨ (*a* ∧ *c*) and *a* ∨ (*b* ∧ *c*) = (*a* ∨ *b*) ∧ (*a* ∨ *c*).

[*B*_{3}]: **Identity Laws:**

*a* ∨ 0 = *a* and *a* ∧ *I* = *a*.

[*B*_{4}]: **Complement Laws:**

*a* ∨ *a*′ = *I* and *a* ∧ *a*′ = 0.

We shall call 0 as zero element, ...