BOOLEAN ALGEBRA
INTRODUCTION
Mathematics is a kind of
shorthand that enables one to present a physical process on paper, with symbols
that may manipulate in order to begin a better understanding of the process. It
is therefore a tool that aids understanding. The familiar kind of Algebra is a
branch of mathematics that deals with variables like x, y, z etc. and
represents combinations and polynomials. In this Algebra the variables can
assume any value of an infinite set of values. This algebra is performed by
analog operational amplifiers. In the case when the variables can assume only
two possible values that is for binary variables George Boole evolved another
set of Algebra in 1854. This algebra has come to be known as Boolean Algebra
after the named of the inventor.
The switching network in a
computer can be very complex, and the logic of a complicated program can be
very difficult to analyze. Boolean Algebra provides a systematic method of
representation and analysis.
A single illustration will show why mathematical logic
is important in the design of computers. If an electronic switch is open, we
shall assign the value 0, and if it is closed, we shall assign the value 1.
These two states of switch will correspond to the values of false (0) and true
(1) in symbolic logic of Boolean Algebra. Two switches which operate
independently such that there is current at the output line C only if both
switches A and B are closed can be described by Boolean statement A AND B. A
circuit in which there will be current at C if either A or B is closed is
described by the Boolean statement A OR B. The result from these circuits can
be described by examining the effect on output C. The work of Boole and others
in the mathematics of symbolic logic thus provided an analytical basis for
Computer design.TRUTH TABLE
In the field of numerical
mathematics, theorems may be proved and operations defined by the process of
enumeration of all possible solutions. Enumeration is particularly well suited
to Boolean Algebra since it involves only two values 1 and 0 and the required
enumeration of all possible solutions is within manageable limits. A truth
table is an enumeration of all solutions of Boolean expression or function.
BASIC BOOLEAN OPERATORS
NOT, AND and OR are three primary
and EXOR, NAND and NOR are three secondary operators of Boolean Algebra. The
AND operation of two binary variables A and B with result C is defined by the
truth table of AND.
Truth Table AND
A

B

C = A AND B

0

0

0

1

0

0

0

1

0

1

1

1

From this table one can see that
the output of an AND operation is 1, if and only if, both the inputs are 1’s or
in general case, the output is 1 when all the inputs are 1’s.
AND
is a logical binary operator and can also be represented by two switches. In
the electrical theory, AND can be modeled by current flow controlled by two
switches. In the circuit, current will flow only when both the switches are
closed. Such a condition is realized by having two switches in series.
Second basic operation of Boolean Algebra is OR.
Truth Table OR
A

B

C = A OR B

0

0

0

1

0

1

0

1

1

1

1

1

In Boolean Algebra, the ‘+’ sign
indicates an OR operation. In OR, in the case of both the operands being 1 the
output is 1. An expression formed with the OR operator is true if either of the
given proposition is true or if all the proposition are true.
Applied
to electrical engineering, the OR operation can be represented as current
controlled by switches placed in parallel. Current flow occurs when either of
the two switches or both the switches are closed.
The third Boolean operation
called NOT or negation is an inversion operation. It is the only binary
operation in Boolean Algebra and is performed on a single bit at a time. In
this operation a sequence of bits is inverted by replacing 0’s by 1’s and
viceversa. The inverse of a sequence A is denoted by A’.
Truth Table NOT
A

A’

0

1

1

0

Exclusive OR or the EXOR
operator defines a relationship between two Boolean variables according to
which an expression formed with ExclusiveOR is true if either one of the two
propositions is true, but is false when both the propositions are true (i.e.
they are mutually exclusive).
Truth table EXOR
A

B

C = A Å B

0

0

0

1

0

1

0

1

1

1

1

0

With regard to switches, it
requires four switches performing a combination of three primary operations to
implement EXOR. It can be written
as ( A AND B’) OR (A’ AND B)
NAND is another secondary Boolean
operator, which can be thought of as a series combination of AND followed by
NOT operation. An expression with the NAND operator is true if either one of
the two propositions is false or both of
them are false. The expression is written as A NAND B.
Truth Table NAND
A

B

C = A NAND B

0

0

1

1

0

1

0

1

1

1

1

0

NOR is the last of the secondary Boolean operators. It is a series combination of OR followed by NOT. An expression formed by NOR is true if and only if both the constituent propositions are false. The expression is written as A NOR B.
Truth Table NOR
A

B

C = A NOR B

0

0

1

1

0

0

0

1

0

1

1

0

In terms of switches, the configuration is schematically
similar to AND operator.
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