54Logic gates

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LOGIC GATES
▢ Analogue and Digital Signals:

Table 1: Analogue vs Digital

Term
Meaning
Example
Analogue signal
Continuously varying voltage/current signal
Sinusoidally varying alternating voltage
Analogue circuit
Circuit associated with analogue signal
AC signal circuit
Digital signal
Signal having only two discrete values
Square wave
Digital circuit
Electronic circuit handling only digital signal
Logic circuit
▢ Logic Gate:
❖ Definition: Digital circuit with one or more input signals but only one output signal
❖ Gate: Digital switch that can be closed/opened to pass or stop digital signal
❖ Logic Gate Condition: Gate opens to pass digital signal only when required logical condition is satisfied

Table 1: Binary States

Binary digit
Meaning
1
ON, true, yes, closed, high
0
OFF, false, no, open, low
❖ Truth Table: Table showing all possible input-output combinations of a logic gate
▢ Boolean Algebra:
❖ Definition: Algebra based on logical states used to describe logic circuits
❖ Boolean Variables: Logical statements represented by 1 and 0
❖ Fundamental Laws:
  • •
    OR law
  • •
    AND law
  • •
    NOT law

Table 1: Basic Boolean Operations

Law
Symbol
Meaning
OR
+
\(A+B=Y\): output true if A or B or both are true
AND
.
\(A.B=Y\): output true only if both A and B are true
NOT
bar
\(\bar A=Y\): output is complement of input
▢ Boolean Algebra Laws:

Table 1: Basic Laws

Law
Formula
Commutative law of OR
\(A+B=B+A\)
Commutative law of AND
\(A.B=B.A\)
Associative law of OR
\(A+(B+C)=(A+B)+C\)
Associative law of AND
\(A.(B.C)=(A.B).C\)
Distributive law
\(A.(B+C)=A.B+A.C\)
OR identity
\(A+0=A\)
OR domination
\(A+1=1\)
OR idempotent
\(A+A=A\)
OR complement
\(A+\bar A=1\)
AND zero
\(A.0=0\)
AND identity
\(A.1=A\)
AND idempotent
\(A.A=A\)
AND complement
\(A.\bar A=0\)
Double complement
\(\bar{\bar A}=A\)
Absorption law
\(A+A.B=A\)
Absorption law
\(A.(A+B)=A\)
Special identity
\(\bar A+A.B=\bar A+B\)
Special identity
\(A.(\bar A+B)=A.B\)
▢ De Morgan's Theorem:

Table 1: De Morgan Laws

Statement
Formula
Complement of sum = product of complements
\(\overline{A+B}=\bar A.\bar B\)
Complement of product = sum of complements
\(\overline{A.B}=\bar A+\bar B\)
▢ Basic Logic Gates:
❖ Types:
  • •
    OR gate
  • •
    AND gate
  • •
    NOT gate
▢ OR Gate:
❖ Definition: Logic gate with two or more inputs and one output; output is 1 if any input is 1
❖ Boolean Expression: \(Y=A+B\)
❖ Switching Equivalent: Parallel switching circuit

Table 1: OR Gate Truth Table

A
B
Y
0
0
0
0
1
1
1
0
1
1
1
1
▢ AND Gate:
❖ Definition: Logic gate with two or more inputs and one output; output is 1 only if all inputs are 1
❖ Boolean Expression: \(Y=A.B\)
❖ Switching Equivalent: Series switching circuit

Table 1: AND Gate Truth Table

A
B
Y
0
0
0
0
1
0
1
0
0
1
1
1
▢ NOT Gate:
❖ Definition: Logic gate with one input and one output; output is complement of input
❖ Boolean Expression: \(Y=\bar A\)
❖ Also Called: Inverter
❖ Input Terminal: Only one input terminal

Table 1: NOT Gate Truth Table

A
Y
0
1
1
0
▢ Combination Gates:
❖ Types:
  • •
    NOR gate
  • •
    NAND gate
  • •
    XOR gate
  • •
    XNOR gate
▢ NOR Gate:
❖ Full Form: NOT + OR
❖ Definition: Output of OR gate is connected to input of NOT gate
❖ Boolean Expression: \(Y=\overline{A+B}\)
❖ Also Called: Inverter gate

Table 1: NOR Gate Truth Table

A
B
Y
0
0
1
0
1
0
1
0
0
1
1
0
❖ Universal Gate: Yes
▢ NAND Gate:
❖ Full Form: NOT + AND
❖ Definition: Output of AND gate is connected to input of NOT gate
❖ Boolean Expression: \(Y=\overline{A.B}\)

Table 1: NAND Gate Truth Table

A
B
Y
0
0
1
0
1
1
1
0
1
1
1
0
❖ Universal Gate: Yes
▢ XOR Gate:
❖ Full Form: Exclusive OR gate
❖ Definition: Output is 1 only when inputs are different
❖ Boolean Expression: \(Y=A.\bar B+\bar A.B\)
❖ Alternative Expression: \(Y=(A+B).\overline{A.B}\)

Table 1: XOR Gate Truth Table

A
B
Y
0
0
0
0
1
1
1
0
1
1
1
0
▢ XNOR Gate:
❖ Full Form: Exclusive NOR gate
❖ Definition: Output is 1 only when inputs are same
❖ Boolean Expression: \(Y=A.B+\bar A.\bar B\)
❖ Relation: XNOR is inverse of XOR gate

Table 1: XNOR Gate Truth Table

A
B
Y
0
0
1
0
1
0
1
0
0
1
1
1
▢ Universal Gates:

Table 1: Universal Gate Facts

Fact
Answer
Universal gates
NAND and NOR
Building blocks of digital circuit
NAND and NOR
Reason
All other gates can be made using NAND or NOR
Digital circuit construction
Can be made by repeated use of NAND gates
NOR gate
Also called inverter gate
▢ Gate Switching Equivalents:

Table 1: Switching Equivalents

Gate
Equivalent circuit
AND
Series switching circuit
OR
Parallel switching circuit
NOT
One input terminal
▢ Read and Digest:

Table 1: Important Points

Fact
Answer
Binary system
Number system using only 0 and 1
Boolean expression
Obeys commutative, associative and distributive laws
OR gate
Output 1 if any input is 1
AND gate
Output 1 only if all inputs are 1
NOT gate
Output is complement of input
NAND gate
AND followed by NOT
NOR gate
OR followed by NOT
XOR gate
Output 1 when inputs are unequal
XNOR gate
Output 1 when inputs are equal
NAND and NOR
Universal gates
NAND and NOR
Building blocks of digital circuit
Digital signal
Only two discrete values
Truth table
All input-output combinations
▢ Objective Answer Key:

Table 1: Logic Gates MCQ Answers

Q
Ans
1
a
2
a
3
b
4
c
5
a
6
b
7
b
8
b
9
d
10
c
11
a
12
c
13
a
14
c
15
b
16
c
▢ High-Yield Recall:

Table 1: Logic Gates One-Liners

Fact
Answer
Analogue signal
Continuously varying signal
Digital signal
Two discrete values
Binary digits
0 and 1
Logic gate
One/more inputs, one output
Truth table
Input-output possibility table
OR expression
\(Y=A+B\)
AND expression
\(Y=A.B\)
NOT expression
\(Y=\bar A\)
NOR expression
\(Y=\overline{A+B}\)
NAND expression
\(Y=\overline{A.B}\)
XOR expression
\(Y=A.\bar B+\bar A.B\)
XNOR expression
\(Y=A.B+\bar A.\bar B\)
AND equivalent
Series switching
OR equivalent
Parallel switching
Universal gates
NAND and NOR
De Morgan 1
\(\overline{A+B}=\bar A.\bar B\)
De Morgan 2
\(\overline{A.B}=\bar A+\bar B\)
AND output 1
Only when all inputs are 1
OR output 0
Only when all inputs are 0
XOR output 1
Inputs different
XNOR output 1
Inputs same