**Chapter - 8**

**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 = 1Thus 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.

1.0 = 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 = 1The 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+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 Law2. 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**

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

Your blog explains algebra in a very good manner, Algebra is the most important and simple topic in mathematics and I am here to share simple and clear definition of algebra that is ,Its a branch of mathematics that substitutes letters in place of numbers means letters represent numbers.

ReplyDeleteyou are correct.. but listen bull shit. do you really know what are you writing ? huh.. :/ idiot people :P just say that this blog is good but there are no solved questions given for clear understanding.. !!

Deleteyou must give the solved questions for some more understanding.. :)

ReplyDelete