Basic Logic Gates
Basic Operators:-
In Digital
logic there are three basic operators, the AND, the OR and the NOT. These three
operators are the very basis for a digital circuit. In fact, almost everything
your computer does can be described in terms of these three operations.
Fortunately, these operations are not difficult to understand, as their
meanings resemble the meanings of the words as used in every day language.
AND
The symbol for
the AND operation and the mathematical expression using AND is as shown in the below .
Y=A.B
The out for the
AND gate is 1 only if both the inputs are (A&B) are 1. Otherwise, the value
is 0.
OR
The symbol for the OR operation and mathematical expression using OR looks like this.
The value of an OR expression is 1 when at
least of of the input value is 1, and 0 otherwise. That is, the above
expression equals 1 if either A or B is 1. The truth table for the OR
operation is as follows.
NOT
NOT is the simplest operation. AND and OR are
binary operations, since they require two values as input. NOT is a unary
operation, and looks like this.
The value of a NOT expression is the opposite
value of the input value.
NAND
and NOR
If the AND, OR
and NOT operators are combined, then the NOR and NAND can be created:
A
NAND B is . This
is the inverted output of the AND gate
A
NAND B looks like this.This
is the inverted output of the AND gate
A
NOR B looks like this. This is just the inverted output of
an OR gate.
XOR
and XNOR
Two other important gates are the exclusive-OR
and exclusive -NOR operators, XOR and XNOR. This is sometimes denoted by a plus
sign in a circle
A
XOR B is 
. This is true only if exactly one of
the inputs is one.
Associativity
Distributivity
Commutativity
. This is true only if exactly one of
the inputs is one.
A
XNOR B is 
. This is the inverted output of an
XOR gate: it is only true if both input are the same.
. This is the inverted output of an
XOR gate: it is only true if both input are the same.
XOR represents
a modulo-2 addition, which means that if you add 1 to 1, you wrap around back
to 0. This is very useful function in digital electronics, but it is not an
important concept in Boolean algebra.
Formal
Mathematical Operators
In the field of
logic, which is part of discrete mathematics, there is an alternative notation
to the addition/multiplication.Unfortunately,
computer science, engineering and mathematics seem unable to establish a
consensus, so we are stuck with both forms of notation. Other books, and
especially those that deal more with pure logic or discrete mathematics may
have various notations, so if other books are consulted, then the other
notation needs to be known. As this is an engineering book, we will not use
this notation.
Boolean
Algebra Laws
Boolean
Algebra, like regular algebra, has certain rules. These rules are
Associativity, Distributivity, Commutativity and De Morgan's Laws.
Associativity, Commutativity and Distributivity only apply to the AND and OR
operators. Some of these laws may seem trivial because you are so used to them.
However, when Boolean algebra was created with its different rules, every axiom
we take for granted in "normal" algebra no longer was guaranteed to
apply. These laws have been proved to hold under Boolean algebra
Associativity
Associativity is the property of algebra that
the order of evaluation of the terms is immaterial.
Distributivity
Distibutivity is the property that an
operator can be applied to each of the terms within the brackets.
Commutativity
Commutativity is the property that order of
application of an operator is immaterial.
De
Morgan's Law
De Morgan's Law is a consequence of the fact
that the NOT or negation operator is not distributive.
De Morgan's laws (named after Augustus De Morgan, 1806–1871) tell us
that a NAND gate gives the same output as an OR gate with inputs complemented
and a NOR gate gives the same output as an AND gate with outputs complemented.
These complemented-input gates are also known as bubbled gates because of the
way that they are indicated on a symbol, i.e., by including a small 'bubble' on
each input, in the same fashion that circles are drawn on the outputs of the
NOT, NAND and NOR gates.
De Morgan's laws are
the most useful while simplifying a boolean expression. An easy way to remember
these laws is "Change the sign, break the line
Categories:
Digital Electronics
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