lecture_binary_logic_and_logic_gates.ppt

Based on slides by:Charles Kime & Thomas Kaminski
© 2004 Pearson Education, Inc.
ECE/CS 352: Digital System Fundamentals
Lecture 4 – Binary Logic
and Logic Gates

Chapter 1 2
Outline
 Binary Logic and Variables
 Logical Operations
 Truth Tables
 Logic Implementation
 Logic Gates

Chapter 1 3
Binary Logic and Gates
 Binary variables take on one of two values.
 Logical operators operate on binary values and
binary variables.
 Basic logical operators are the logic functions
AND, OR and NOT.
 Logic gates implement logic functions.
 Boolean Algebra: a useful mathematical system
for specifying and transforming logic functions.
 We study Boolean algebra as foundation for
designing and analyzing digital systems!

Chapter 1 4
Binary Variables
 Recall that the two binary values have
different names:
• True/False
• On/Off
• Yes/No
• 1/0
 We use 1 and 0 to denote the two values.
 Variable identifier examples:
• A, B, y, z, or X1 for now
• RESET, START_IT, or ADD1 later

Chapter 1 5
Logical Operations
 The three basic logical operations are:
• AND
• OR
• NOT
 AND is denoted by a dot (·).
 OR is denoted by a plus (+).
 NOT is denoted by an overbar ( ¯ ), a
single quote mark (') after, or (~) before
the variable.

Chapter 1 6
 Examples:
• is read “Y is equal to A AND B.”
• is read “z is equal to x OR y.”
• is read “X is equal to NOT A.”
Notation Examples
 Note: The statement:
1 + 1 = 2 (read “one plus one equals two”)
is not the same as
1 + 1 = 1 (read “1 or 1 equals 1”).
 B
A
Y 
y
x
z 

A
X 

Chapter 1 7
Operator Definitions
 Operations are defined on the values
"0" and "1" for each operator:
AND
0 · 0 = 0
0 · 1 = 0
1 · 0 = 0
1 · 1 = 1
OR
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
NOT
1
0
0
1

Chapter 1 8
0
1
1
0
X
NOT
X
Z
Truth Tables
 Truth table  a tabular listing of the values of a
function for all possible combinations of values on its
arguments
 Example: Truth tables for the basic logic operations:
1
1
1
0
0
1
0
1
0
0
0
0
Z = X·Y
Y
X
AND OR
X Y Z = X+Y
0 0 0
0 1 1
1 0 1
1 1 1

Chapter 1 9
 Using Switches
• For inputs:
 logic 1 is switch closed
 logic 0 is switch open
• For outputs:
 logic 1 is light on
 logic 0 is light off.
• NOT uses a switch such
that:
 logic 1 is switch open
 logic 0 is switch closed
Logic Function Implementation
Switches in series => AND
Switches in parallel => OR
C
Normally-closed switch => NOT

Chapter 1
 Example: Logic Using Switches
 Light is on (L = 1) for
L(A, B, C, D) =
and off (L = 0), otherwise.
 Useful model for relay circuits and for CMOS
gate circuits, the foundation of current digital
logic technology
Logic Function Implementation (Continued)
B
A
D
C
A ((B C') + D) = A B C' + A D

Chapter 1
Logic Gates
 In the earliest computers, switches were opened
and closed by magnetic fields produced by
energizing coils in relays. The switches in turn
opened and closed the current paths.
 Later, vacuum tubes that open and close
current paths electronically replaced relays.
 Today, transistors are used as electronic
switches that open and close current paths.

Chapter 1
Logic Gates (continued)
 Implementation of logic gates with transistors (See
Reading Supplement  CMOS Circuits)
 Transistor or tube implementations of logic functions are
called logic gates or just gates
 Transistor gate circuits can be modeled by switch circuits
•
F
+V
X
Y
+V
X
+V
X
Y
•
•
•
•
•
• •
•
• •
•
•
(a) NOR
G = X +Y
(b) NAND (c) NOT
X .Y
X
•
•
•

Chapter 1
(b) Timing diagram
X 0 0 1 1
Y 0 1 0 1
X · Y
(AND) 0 0 0 1
X 1 Y
(OR) 0 1 1 1
(NOT) X 1 1 0 0
(a) Graphic symbols
OR gate
X
Y
Z 5 X 1 Y
X
Y
Z 5 X · Y
AND gate
X Z 5 X
Logic Gate Symbols and Behavior
 Logic gates have special symbols:
 And waveform behavior in time as follows:

Chapter 1
Logic Diagrams and Expressions
 Boolean equations, truth tables and logic diagrams describe the same function!
 Truth tables are unique; expressions and logic diagrams are not. This gives
flexibility in implementing functions.
X
Y F
Z
Logic Diagram
Equation
Z
Y
X
F 

Truth Table
1
1 1 1
1
1 1 0
1
1 0 1
1
1 0 0
0
0 1 1
0
0 1 0
1
0 0 1
0
0 0 0
X Y Z Z
Y
X
F 



Chapter 1
Summary
 Binary Logic and Variables
 Logical Operations
 Truth Tables
 Logic Implementation
 Logic Gates

Chapter 1
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 © 2004 by Pearson Education,Inc. All rights reserved.
 The following terms of use apply in addition to the standard
Pearson Education Legal Notice.
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presentations and handouts only to instructors adopting Logic and
Computer Design Fundamentals as the course text.
 Permission is granted to the instructors adopting the book to post
these materials on a protected website or protected ftp site in
original or modified form. All other website or ftp postings,
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 You may not remove or in any way alter this Terms of Use notice
or any trademark, copyright, or other proprietary notice,
including the copyright watermark on each slide.
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