## CHAPTER # 8 – BOOLEAN ALGEBRA

**Boolean Algebra**

The Boolean algebra was developed by the English mathematician George Boole; it deals with statements in mathematical logic, and puts them in the form of algebraic equations. The Boolean algebra was further developed by the modern American mathematician Claude Shannon, in order to apply it to computers. The basic techniques described by Shannon were adopted almost universally for the design and analysis of switching circuits. Because of the analogous relationship between the actions of relays, and of modern electronic circuits, the same techniques which were developed for the design of relay circuits are still being used in the design of modern high speed computers. Thus the Boolean algebra founds its applications in modern computers after almost one hundred years of its discovery.

Boolean algebra provides an economical and straightforward approach to the design of relay and other types of switching circuits. Just as an ordinary algebraic expression may be simplified by means of the basic theorems, the expression describing a given switching circuit network may also be reduced or simplified using Boolean algebra.

Boolean algebra is used in designing of logic circuits inside the computer. These circuits perform different types of logical operations. Thus, Boolean algebra is also known as logical algebra or switching algebra. The mathematical expressions of the Boolean algebra are called Boolean expressions. Boolean algebra describes the Boolean expressions used in the logic circuits. The Boolean expressions are simplified by means of basic theorems. The expressions that describe the logic circuits are also simplified by using Boolean theorems.

Boolean algebra is now being used extensively in designing the circuitry used in computers. In short, knowledge of Boolean-algebra is must in the computing field.

**Definitions**

*Constants*

Boolean algebra uses binary values 0 and 1 as Boolean constants.

*Variable*

The variables used in the Boolean algebra are represented by letters such as A, B, C, x, y, z etc, with each variable having one of two and only two distinct possible values 0 and 1.

*Truth Table*

It is defined as systematic listing of the values for the dependent variable in terms of all the possible values of independent variable. It can also be defined as a table representing the condition of input and output circuit involving two or more variables. In a binary system, there is 2(n) number of combinations, where n is he number of variables being used for e.g. each combination of the value of x and y, there is value of z specified by the definition. These definitions may listed in compact form using “Truth Tables”. Therefore a truth table is able of all possible combinations of the variables.

*AND Operation*

In Boolean algebra AND operator is represented by a dot or by the absence of any symbol between the two variables and is used for logical multiplication. For example A.B = X or AB = X.

Thus X is 1 if both A and B are equal to 1 otherwise X will be 0 if either or both A and B are 0 i.e.

1.1 = 1

1.0 = 0

0.1 = 0

0.0 = 0

*OR Operation*

OR operation is represented by a plus sign between two variables. In Boolean algebra OR is used for logical addition. For example A+B = X.

The resulting variable X assumes the value 0 only when both A nd B are 0, otherwise X will be 1 if either or both of A and B are 1 i.e.

1+1 = 1

1+0 = 1

0+1 = 1

0+0 = 0

**Laws of Boolean Algebra**

As in other areas of mathematics, there are certain well-defined rules and laws that must be followed in order to properly apply Boolean algebra. There are three basic laws of Boolean algebra; these are the same as ordinary algebra.

1. Commutative Law

2. Associative Law

3. Distributive Law

*1. Commutative Law*

It is defined as the law of addition for two variables and it is written as:

A + B = B + A

This law states that the order in which the variables are added makes no difference. Remember that in Boolean algebra addition and OR operation are same. It is also defined as the law of multiplication for two variables and it is written as:

A.B = B.A

*2. Associative Law*

The associative law of addition is written as follows for three variables:

A + (B + C) = (A + B) + C

This law states that when ORing more than two variables, the result is the same regardless of the grouping of the variables.

The associative law of multiplication is written as follows for three variables.

A(BC) = (AB)C

This law states that it makes no difference in what order the variables are grouped when ANDing more than two variables.

*3. Distributive Laws*

The distributive law is written for three variables is as follows:

A(B+C) = AB + AC

This law states that ORing two or more variables and then ANDin the result with a single variable is equivalent to ANDing the single variable with each of the two or more variables and then ORing the products. The distributive law also expresses the process of factoring in which the common variable A is factored out of the product terms. For example:

AB + AC = A (B + C)