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Logic Gates

The Logic gates are electronic circuits that can be used to implement the most

elementary logic expressions, also known as Boolean expressions. The logic gate is

the most basic building block of combinational logic. There are three basic logic

gates, namely the OR gate, the A! gate and the OT gate. Other logic gates that

are deri"ed from these basic gates are the A! gate, the OR gate, the

#\$%L&'()#OR gate and the #\$%L&'()#*OR gate.

Positive and Negative Logic

The binary "ariables, as we know, can ha"e either of the two states, i.e. the logic +-

state or the logic +- state. These logic states in digital systems such as computers, for

instance, are represented by two different "oltage le"els or two different current

le"els. (f the more positi"e of the two "oltage or current le"els represents a logic +-

and the less positi"e of the two le"els represents a logic +-, then the logic system is

referred to as a positive logic system. (f the more positi"e of the two "oltage or

current le"els represents a logic +- and the less positi"e of the two le"els represents a

logic +-, then the logic system is referred to as a negative logic system. The

following examples further illustrate this concept. (f the two "oltage le"els are )

and /0 ), then in the positi"e logic system the ) represents a logic +- and the /0 )

represents a logic +-. (n the negati"e logic system, ) represents a logic +- and /0

) represents a logic +-. (f the two "oltage le"els are ) and 10 ), then in the

positi"e logic system the ) represents a logic +- and the 10 ) represents a logic

+-. (n the negati"e logic system, ) represents a logic +- and 10 ) represents a

logic +-. (t is interesting to note, as we will disco"er in the latter part of the chapter,

that a positi"e OR is a negati"e A!. That is, OR gate hardware in the positi"e logic

system beha"es like an A! gate in the negati"e logic system. The re"erse is also

true. 'imilarly, a positi"e OR is a negati"e A!, and "ice "ersa.

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OR Gate

An OR gate performs an ORing operation on two or more than two logic "ariables.

The OR operation on two independent logic "ariables A and B is written as 2 3 A/B

and reads as 2 e4uals A OR B and not as A plus B. An OR gate is a logic circuit with

two or more inputs and one output. The output of an OR gate is LO5 only when all

of its inputs are LO5. 6or all other possible input combinations, the output is 7(87.

This statement when interpreted for a positi"e logic system means the following.

The output of an OR gate is a logic +- only when all of its inputs are at logic +-. 6or

all other possible input combinations, the output is a logic +-. 6igure below shows

the circuit symbol and the truth table of a two*input OR gate. The operation of a two*

input OR gate is explained by the logic expression9

2 3 A/B

A B 2

As an illustration, if we ha"e four logic "ariables and we want to know the logical

output of :A/ B/% /!;, then it would be the output of a four*input OR gate with A,

B, % and ! as its inputs. 23A/B 6igures :a; and :b; show the circuit symbol of

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three*input and four*input OR gates. 6igure :c; shows the truth table of a three*input

OR gate. Logic expressions explaining the functioning of threei nput and four*input

OR gates are 2 3 A/B/% and 2 3 A/B/% /!.

Q/Ho ould you hardare!implement a "our!input O# gate using to!input O#

gates only\$

=

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Example : Draw the output waveform for the OR gate and the given pulsed input

waveforms of Fig

!ND Gate

An A! gate is a logic circuit ha"ing two or more inputs and one output. The output

of an A! gate is 7(87 only when all of its inputs are in the 7(87 state. (n all other

cases, the output is LO5. 5hen interpreted for a positi"e logic system, this means

that the output of the A! gate is a logic +- only when all of its inputs are in logic

+- state. (n all other cases, the output is logic +-. The logic symbol and truth table of

a two*input A! gate are shown below The A! operation on two independent

logic "ariables A and B is written as 2 3 A.B and reads as 2 e4uals A A! B and not

as A multiplied by B. 7ere, A and B are input logic "ariables and 2 is the output. An

A! gate performs an A!ing operation9

>

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0

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? for a two*input A! gate, 2 3 [email protected]

? for a three*input A! gate, 2 3 A.B.%@

? for a four*input A! gate, 2 3 A.B.%.!.

(f we interpret the basic definition of OR and A! gates for a negati"e logic system,

we ha"e an interesting obser"ation. 5e find that an OR gate in a positi"e logic

system is an A! gate in a negati"e logic system. Also, a positi"e A! is a negati"e

OR.

Example :

Sho the logic arrangement "or implementing a "our!input AN% gate using to!input

AN% gates only.

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NO" Gate

A OT gate is a one*input, one*output logic circuit whose output is always the

complement of the input. That is, a LO5 input produces a 7(87 output, and "ice

"ersa. 5hen interpreted for a positi"e logic system, a logic +- at the input produces a

logic +- at the output, and "ice "ersa. (t is also known as a +complementing circuit- or

an +in"erting circuit-.

The OT operation on a logic "ariable \$ is denoted as or . That is, if \$ is the

input to a OT circuit, then its output 2 is gi"en by 2 3 or and reads as 2 e4uals

OT \$. Thus, if \$ 3 , 2 3 and if \$ 3 , 2 3 .

# \$For the logic circuit arrangements of Figs %a& and %'&( draw the output

waveform

E)*L+,-.E/OR Gate

The #\$%L&'()#*OR gate, commonly written as #\$*OR gate, is a two*input, one*

output gate. 6igures :a; and :b; respecti"ely show the logic symbol and truth table of

a two*input #\$*OR gate. As can be seen from the truth table, the output of an #\$*OR

gate is a logic +- when the inputs are unlike and a logic +- when the inputs are like.

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Although #\$*OR gates are a"ailable in integrated circuit form only as two*input

gates, unlike other gates which are a"ailable in multiple inputs also, multiple*input

#\$*OR logic functions can be implemented using more than one two*input gates.

The truth table of a multiple*input #\$*OR function can be expressed as follows. The

output of a multiple*input #\$*OR logic function is a logic +- when the number of s

in the input se4uence is odd and a logic +- when the number of s in the input

se4uence is e"en, including Cero. That is, an all s input se4uence also produces a

logic +- at the output. 6igure:c; shows the truth table of a four*input #\$*OR

function. The output of a two*input #\$*OR gate is expressed by

2 3 AB 3 B/A

D

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#: 0ow do 1ou implement three/input and four/input )/OR logic functions with

the help of two/input E)/OR gates2#: 0ow can 1ou implement a NO" circuit using a two/input E)/OR gate2

N!ND Gate

A! stands for OT A!. An A! gate followed by a OT circuit makes it a

A! gate E6ig. :a;F. 6igure :b; shows the circuit symbol of a two*input A!

gate. The truth table of a A! gate is obtained from the truth table of an A! gate

by complementing the output entries E6ig. :c;F. The output of a A! gate is a logic

G

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+- when all its inputs are a logic +-. 6or all other input combinations, the output is a

logic +-. A! gate operation is logically expressed as

2 3

(n general, the Boolean expression for a A! gate with more than two inputs can

be written as

2 3

NOR Gate

OR stands for OT OR. An OR gate followed by a OT circuit makes it a OR

gate E6ig. :a;F. The truth table of a OR gate is obtained from the truth table of an

OR gate by complementing the output entries. The output of a OR gate is a logic +-

when all its inputs are logic +-. 6or all other input combinations, the output is a logic

+-. The output of a two*input OR gate is logically expressed as

2 3

(n general, the Boolean expression for a OR gate with more than two inputs can be

written as

2 3

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E)*L+,-.E/NOR Gate

#\$%L&'()#*OR :commonly written as #\$*OR; means OT of #\$*OR, i.e. the

logic gate that we get by complementing the output of an #\$*OR gate. The truth

table of an #\$*OR gate is obtained from the truth table of an #\$*OR gate by

complementing the output entries. Logically, The output of a two*input #\$*OR gate

is a logic +- when the inputs are like and a logic +- when they are unlike. (n general,the output of a multiple*input #\$*OR logic function is a logic +- when the number

of s in the input se4uence is odd and a logic +- when the number of s in the input

se4uence is e"en including Cero. That is, an all s input se4uence also produces a

logic +- at the output.

2 3 3 AB/

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03 :

,how the logic arrangements for implementing:

%a& a four/input N!ND gate using two/input !ND gates and NO" gates4

%'& a three/input N!ND gate using two/input N!ND gates4

%c& a NO" circuit using a two/input N!ND gate4

%d& a NO" circuit using a two/input NOR gate4

%e& a NO" circuit using a two/input E)/NOR gate

03:

0ow do 1ou implement a three/input E)/NOR function using onl1 two/input E)/

NOR gates2