ECE 331 – Digital System Design
Basic Logic Operations,
Boolean Expressions and Truth Tables,
and
Standard Logic Gates
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
Basic Logic Operations
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Basic Logic Operations



AND

Logical multiplication (product)

A . B = AB = “A and B”
OR

Logical addition (sum)

A + B = “A or B”
Order of Precedence:
1. NOT
2. AND
3. OR
- can be modified by parenthesis
NOT
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
Logical complement

A' = “not A”
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Truth Tables
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AND
OR
NOT
NOT
A
B
A.B
A+B
A'
B'
0
0
0
0
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
1
1
1
0
0
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Logic Gates
NOT
A
A'
AND
OR
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Additional Logic Operations


NAND

F = (A . B)'

F = not (A and B)

Compound logic function
Not the same as F = A' . B'
NOR
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
F = (A + B)'

F = not (A or B)

Compound logic function
Not the same as F = A' + B'
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Truth Tables
Exercise:
Derive the Truth Table for the
NAND and NOR gates.
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Logic Gates
shorthand for a NOT gate
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Additional Logic Operations


Exclusive-OR

F = (A.B') + (A'.B)

Output = 1 iff one of the inputs is 1 (but not both).

Inclusive-OR: F = A + B
Exclusive-NOR
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
F = (A.B) + (A'.B')

Output = 1 iff both input are 0 or both inputs are 1.

Also known as the Equivalence function.
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Truth Tables
Exercise:
Derive the Truth Table for the
XOR and XNOR gates.
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Logic Gates
Exclusive-OR
Equivalence
Exclusive-NOR
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Truth Tables
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Truth Tables


Used to describe the functional behavior of a Boolean
expression and/or Logic circuit.
Each row in the truth table represents a unique
combination of the input variables.




For n input variables, there are 2n rows.
The output of the logic function is defined for each row.
Each row is assigned a numerical value, with the rows
listed in ascending order.
The order of the input variables defined in the logic
function is important.
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3-input Truth Table
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#
A
B
C
0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
4
1
0
0
5
1
0
1
6
1
1
0
7
1
1
1
ECE 331 - Digital System Design
F(A,B,C)
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4-input Truth Table
#
A
B
C
D
0
0
0
0
0
1
0
0
0
1
2
0
0
1
0
3
0
0
1
1
4
0
1
0
0
5
0
1
0
1
6
0
1
1
0
12
1
1
0
0
13
1
1
0
1
14
1
1
1
0
15
1
1
1
1
F(A,B,C,D)
…
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Boolean Expressions
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Boolean Expressions


Boolean expressions are composed of

Literals – variables and their complements

Logical operations
Examples

F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C'
literals
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logic operations

F = (A+B+C').(A'+B'+C).(A+B+C)

F = A.B'.C' + A.(B.C' + B'.C)
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Boolean Expressions

Boolean expressions are realized using a
network (or combination) of logic gates.

Each logic gate implements one of the logic
operations in the Boolean expression

Each input to a logic gate represents one of
the literals in the Boolean expression
A
B
logic operations
literals
F
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Boolean Expressions



Boolean expressions are evaluated by

Substituting a 0 or 1 for each literal

Calculating the logical value of the expression
A Truth Table specifies the value of the Boolean
expression for every combination of the
variables in the Boolean expression.
For an n-variable Boolean expression, the truth
n
table has 2 rows (one for each combination).
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Boolean Expressions
Example:
Evaluate the following Boolean expression, for all
combination of inputs, using a Truth table.
F(A,B,C) = A'.B'.C + A.B'.C'
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Boolean Expressions


Two Boolean expressions are equivalent if they
have the same value for each combination of
the variables in the Boolean expression.

F1 = (A + B)'

F2 = A'.B'
How do you prove that two Boolean
expressions are equivalent?
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
Truth table

Boolean Algebra
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Boolean Expressions
Example:
Using a Truth table, prove that the following two
Boolean expressions are equivalent.
F1 = (A + B)'
F2 = A'.B'
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Standard Logic Gates
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Standard Logic Gates


Discrete components used to build logic circuits.

74xx08
AND gate

74xx32
OR gate

74xx04
NOT gate

74xx00
NAND gate

74xx02
NOR gate

74xx86
XOR gate
Logic Families (TTL, LS-TTL, F, HC, …)
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Standard Logic Gates: Data sheets



Data sheets provide essential information:

Logic Function

Truth Table

Pin-out

Electrical Characteristics

Timing Characteristics

Package Description(s)
This information is necessary when building logic
circuits from discrete components.
Each Logic Family has a unique set of characteristics.
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

Logic Families
TTL

Low-Power TTL (“L”)

High-Speed TTL (“H”)

Schottky (“S”)

Low-Power Schottky (“LS”)

Advanced Schottky (“AS”)

Advanced Low-Power Schottky (“ALS”)

Fast (“F”)
CMOS
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
High-Speed CMOS (“HC”)

Advanced CMOS (“AC”)
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Comparison of Logic Families
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Combinational Logic Circuits
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Combinational Logic Circuits

Composed of an interconnected set of logic gates.

Also known as Switching Circuits

Logic circuits can be designed from


Truth tables

Boolean expressions
Logic circuits realized through
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
Interconnection of discrete components

Synthesis from a Hardware Description Language
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Example:
Derive the Boolean expression and draw the
circuit diagram for the following truth table:
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A
B
F(A,B)
0
0
1
0
1
0
1
0
1
1
1
1
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Exercise:
Derive the Boolean expression and draw the
circuit diagram for the following truth table:
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A
B
C
F(A,B,C)
0
0
0
0
0
0
1
1
0
1
0
1
0
1
1
0
1
0
0
1
1
0
1
0
1
1
0
0
1
1
1
1
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Wiring Diagrams
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Standard Logic Gate: Pin-out
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Standard Logic Gate: Pin-out
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Standard Logic Gate: Pin-out
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Example:
Draw the circuit diagram and the wiring diagram
for the following Boolean expression:
F = B'.C + A.B
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Circuit Diagram
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Wiring Diagram
VDD
7404
7408
A
B
C
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7432
F
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