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 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.
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 the 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.
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 and B are 0, otherwise X will be 1 if either or both of A and B are 1 i.e.
The resulting variable X assumes the value 0 only when both A and 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
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
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
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.
Forexample:
AB + AC = A (B + C).
AB + AC = A (B + C).
