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COMPUTER: 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
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
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.
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) |
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