R.M. Dansereau; v.1.0
INTRO. TO COMP. ENG.
CHAPTER III-1
BOOLEAN ALGEBRA
•CHAPTER III
CHAPTER III
BOOLEAN ALGEBRA
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Boolean Algebra: Fundamentals, Operations, and Standard Forms - Prof. Vladimir Goncharoff, Study notes of Electrical and Electronics Engineering
An introduction to boolean algebra, including the fundamental operators not, and, and or, as well as their operations and boolean functions. It also covers the concepts of minterms and maxterms, sum-of-minterms and product-of-maxterms standard forms, and simplification using karnaugh maps.
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INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA
•CHAPTER III
CHAPTER III
BOOLEAN ALGEBRA
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN VALUES
INTRODUCTION
BOOLEAN ALGEBRA
- BOOLEAN VALUES -^
Boolean algebra is a form of algebra that deals with single digit binaryvalues and variables.
-^
Values and variables can indicate some of the following binary pairs ofvalues:•^
ON / OFF
•^
TRUE / FALSE
•^
HIGH / LOW
•^
CLOSED / OPEN
•^
INTRO. TO COMP. ENG.
CHAPTER III-
BOOL. OPERATIONS
BINARY BOOLEAN OPERATORS
BOOLEAN ALGEBRA
- BOOLEAN OPERATIONS
-FUNDAMENTAL OPER.
-^
Below is a table showing all possible Boolean functions
given the two-
inputs
and
F
N
A
B
A
B
F
0
F
1
F
2
F
3
F
4
F
5
F
6
F
7
F
8
F
9
F
10
F
11
F
12
F
13
F
14
F
15
AB
A
B
A
B
AB
A
B
A
B
B
A
A
B
Null
Identity
Inhibition
Implication
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA PRECEDENCE OF OPERATORS
BOOLEAN ALGEBRA
- BOOLEAN OPERATIONS
-FUNDAMENTAL OPER.-BINARY BOOLEAN OPER.
-^
Boolean expressions must be evaluated with the following order of operatorprecedence •^
parentheses
-^
NOT
•^
AND
•^
OR
Example:
F
A C
BD
(^
)^
BC
(^
)E
F
A
C
B D
^
^
^
^
B C
^
^
^
^
^
E
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA
BASIC IDENTITIES
BOOLEAN ALGEBRA
- BOOLEAN OPERATIONS • BOOLEAN ALGEBRA
-PRECEDENCE OF OPER.-FUNCTION EVALUATION
X
X
X
X
X
X
(^
X
X
Y
Y
X
X
⋅^
X
X
⋅^
X
X
⋅^
XY
YX
X YZ
(^
)^
XY(
)Z
X
Y
Z
(^
X
Y
(^
)^
Z
X Y
Z
(^
)^
XY
XZ
X
YZ
X
Y
(^
)^
X
Z
(^
Identity Idempotent LawComplementInvolution Law CommutativityAssociativityDistributivity
X
X
⋅^
X
X
X
X
Absorption Law
X X
Y
(^
)^
X
X
XY
X
Simplification
X X
′^
Y
(^
)^
XY
X
X
′Y
X
Y
DeMorgan
’s Law
XY(
X
′^
Y
X
Y
(^
X
′Y
Consensus Theorem
X
Y
(^
)^
X
′^
Z
(^
)^
Y
Z
(^
XY
X
′Z
YZ
XY
X
′Z
X
Y
(^
)^
X
′^
Z
(^
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA
DUALITY PRINCIPLE
BOOLEAN ALGEBRA
- BOOLEAN ALGEBRA
-PRECEDENCE OF OPER.-FUNCTION EVALUATION-BASIC IDENTITIES
-^
Duality principle: •^
States that a Boolean equation remains valid if we take the dual of theexpressions on both sides of the equals sign.
-^
The dual can be found by interchanging the
AND
and
OR
operators
along with also interchanging the
’s and
’s.
•^
This is evident with the duals in the basic identities.
•^
For instance: DeMorgan
’s Law can be expressed in two forms
X
Y
(^
X
′Y
XY
(^
X
′^
Y
as well as
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA FUNCTION MANIPULATION (2)
BOOLEAN ALGEBRA
- BOOLEAN ALGEBRA
-BASIC IDENTITIES-DUALITY PRINCIPLE-FUNC. MANIPULATION
-^
Example: Simplify the following expression •^
Simplification
F
A
AB
ABC
ABCD
ABCDE
F
A
A B
BC
BCD
BCDE
(^
F
A
B
BC
BCD
BCDE
F
A
B
B C
CD
CDE
(^
F
A
B
C
CD
CDE
F
A
B
C
C D
DE
(^
F
A
B
C
D
DE
F
A
B
C
D
E
INTRO. TO COMP. ENG.
CHAPTER III-
BOOLEAN ALGEBRA FUNCTION MANIPULATION (3)
BOOLEAN ALGEBRA
- BOOLEAN ALGEBRA
-BASIC IDENTITIES-DUALITY PRINCIPLE-FUNC. MANIPULATION
-^
Example: Show that the following equality holds •^
Simplification
A BC
BC
(^
)^
A
B
C
(^
)^
B
C
(^
A BC
BC
(^
)^
A
BC
BC
(^
A
BC
(^
)^
BC
(^
A
B
C
(^
)^
B
C
(^
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS
MINTERMS
BOOLEAN ALGEBRA
- BOOLEAN ALGEBRA • STANDARD FORMS
-SOP AND POS
-^
The following table gives the minterms for a
three-input
system
A
B
C
ABC
ABC
ABC
ABC
ABC
ABC
ABC
ABC
m
2
m
3
m
4
m
1
m
0
m
5
m
6
m
7
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS
SUM OF MINTERMS
BOOLEAN ALGEBRA
- BOOLEAN ALGEBRA • STANDARD FORMS
-SOP AND POS-MINTERMS
-^
Sum-of-minterms
standard form expresses the Boolean or switching
expression in the form of a
sum of products
using
minterms
•^
For instance, the following Boolean expression using mintermscould instead be expressed asor more compactly
F A B C
,^
(^
)^
ABC
ABC
ABC
ABC
F A B C
,^
(^
m
0
m
1
m
4
m
5
F A B C
,^
(^
m 0 1 4 5
,^
,^
(^
one-set 0 1 4 5
,^
,^
(^
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS^ PRODUCT OF MAXTERMS
BOOLEAN ALGEBRA
- STANDARD FORMS
-MINTERMS-SUM OF MINTERMS-MAXTERMS
-^
Product-of-maxterms
standard form expresses the Boolean or switching
expression in the form of
product of sums
using
maxterms
•^
For instance, the following Boolean expression using maxtermscould instead be expressed asor more compactly as
F A B C
,^
(^
)^
A
B
C
(^
)^
A
B
C
(^
)^
A
B
C
(^
F A B C
,^
(^
M
1
M
4
M
7
⋅^
F A B C
,^
(^
M 1 4 7
,^
(^
zero-set 1 4 7
,^
(^
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS
MINTERM AND MAXTERM EXP.
BOOLEAN ALGEBRA
- STANDARD FORMS
-SUM OF MINTERMS-MAXTERMS-PRODUCT OF MAXTERMS
-^
Given an arbitrary Boolean function, such ashow do we form the canonical form for: •^
sum-of-minterms
•^
Expand the Boolean function into a sum of products. Then takeeach term with a missing variable
and
AND
it with
•^
product-of-maxterms
•^
Expand the Boolean function into a product of sums. Then takeeach factor with a missing variable
and
OR
it with
F A B C
,^
(^
)^
AB
B A
C
(^
X
X
X
X
XX
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS
FORMING PROD OF MAXTERMS
BOOLEAN ALGEBRA
- STANDARD FORMS
-PRODUCT OF MAXTERMS-MINTERM & MAXTERM-FORM SUM OF MINTERMS
•^
Example
F A B C
,^
(^
)^
AB
B A
C
(^
AB
AB
BC
A
B
(^
)^
A
B
C
(^
)^
A
B
C
(^
M 2 3 5
,^
(^
A
B
CC
(^
)^
A
B
C
(^
)^
A
B
C
(^
A
B
C
(^
)^
A
B
C
(^
)^
A
B
C
(^
(using distributivity)
A
B
C
F^11001011
Maxterms listed as
0s in Truth Table
INTRO. TO COMP. ENG.
CHAPTER III-
STANDARD FORMS
CONVERTING MIN AND MAX
BOOLEAN ALGEBRA
- STANDARD FORMS
-MINTERM & MAXTERM-SUM OF MINTERMS-PRODUCT OF MAXTERMS
-^
Converting between sum-of-minterms and product-of-maxterms •^
The two are complementary, as seen by the truth tables.
-^
To convert interchange the
and
, then use missing terms.
•^
Example: The example from the previous slidesis re-expressed aswhere the numbers 2, 3, and 5 were missing from the mintermrepresentation.
F A B C
,^
(^
m 0 1 4 6 7