Title: ECE 331
1ECE 331 Digital System Design
- Truth Tables,
- Boolean Expressions,
- and
- Boolean Algebra
- (Lecture 3)
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.
2Logic Functions
- A logic function can be described by a
- Truth table
- Boolean expression (i.e. equation)
- Circuit diagram (aka. Logic Circuit)
- Each can equally describe the logic function.
3 Truth Tables
4Truth Tables
- A truth table defines the value of the output of
a logic function for each combination of the
input variables. - Each row in the truth table corresponds to a
unique combination of the input variables. - For n input variables, there are 2n rows.
- 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.
53-input Truth Table
A B C F(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
3 variables ? 23 8 rows
64-input Truth Table
A B C D F(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
4 variables ? 24 16 rows
7Truth Tables Examples
A B C F1(A,B,C)
0 0 0 0
0 0 1 0
0 1 0 1
0 1 1 1
1 0 0 0
1 0 1 1
1 1 0 0
1 1 1 1
A B C F2(A,B,C)
0 0 0 1
0 0 1 0
0 1 0 1
0 1 1 0
1 0 0 0
1 0 1 1
1 1 0 1
1 1 1 0
8 Boolean Expressions (and Logic Circuits)
9Boolean Expressions
- A Boolean expression is composed of
- Literals variables and their complements
- Logical operators
- Examples
- F1 A.B.C A'.B'.C A.B'.C' A'.B.C'
- F2 (A'BC).(AB'C).(ABC')
- F3 A'.(BC) B.(AC')
Literals highlighted in green Logical operators
highlighted in blue
10Boolean Expressions
- A Boolean expression is evaluated by
- Substituting a 0 or 1 for each literal
- Calculating the logical value of the expression
- A truth table represents the evaluation of a
Boolean expression for all combinations of the
input variables.
11Boolean Expressions Example 1
Simple AND and OR Functions
- Using a truth table, evaluate the following
Boolean expressions - F1(A,B,C) A'.B.C'
- F2(A,B,C) A B' C'
An AND function 1 when all literals 1.
An OR function 1 when any literal 1.
Literal X or X' If X' 1 then X 0 If X' 0
then X 1
An OR function 0 when all literals 0.
12Boolean Expressions Example 1
A B C A' B' C' F1 F2
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
13Boolean Expressions Example 2
More Complex Functions of ANDs and ORs
- Using a truth table, evaluate the following
Boolean expression - F(A,B,C) A'.C B.C' A.B'.C'
An AND term 1 when all literals 1.
An OR function 1 when any term 1.
14Boolean Expressions Example 2
A B C A'C BC' AB'C' F(A,B,C)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
15Boolean Expressions Example 3
More Complex Functions of ANDs and ORs
- Using a truth table, evaluate the following
Boolean expression - F(A,B,C) (AB').(A'C).(AB'C')
An OR term 1 when any literal 1.
An AND function 1 when all terms 1.
16Boolean Expressions Example 3
A B C AB' A'C AB'C' F(A,B,C)
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
17Logic Circuits
- A Boolean expression is realized using a network
of logic gates, known as a logic circuit or a
circuit diagram, where - Each logic gate represents a logical operator
- Each input to a logic gate represents a literal
Circuit Diagram
18(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 are realized through
- Interconnection of discrete components
- Synthesis from a Hardware Description Language
19Logic Circuit Example 1
- Given the following truth table,
- 1. Derive a Boolean expression
- 2. Draw the corresponding circuit diagram
A B F(A,B)
0 0 1
0 1 0
1 0 1
1 1 1
20Logic Circuit Example 2
- Given the following truth table,
- 1. Derive a Boolean expression
- 2. Draw the corresponding circuit diagram
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
21Equivalency of Boolean Expressions
- Two Boolean expressions are equivalent iff they
have the same value for each combination of the
variables in the Boolean expression. - How do you prove that two Boolean expressions are
equivalent? - Truth table
- Boolean Algebra
22Equivalence Example
- Using a Truth table, prove that the following two
Boolean expressions are equivalent. - F1(A,B) A'.B A.B'
- F2(A,B) (A'.B' A.B)'
23Equivalence Example
A B A'B AB' A'B' AB F1 F2
0 0
0 1
1 0
1 1
24 Boolean Algebra
25Boolean Algebra
- George Boole developed an algebraic description
for processes involving logical thought and
reasoning. - Became known as Boolean Algebra
- Claude Shannon later demonstrated that Boolean
Algebra could be used to describe switching
circuits. - Switching circuits are circuits built from
devices that switch between two states (e.g. 0
and 1). - Switching Algebra is a special case of Boolean
Algebra in which all variables take on just two
distinct values - Boolean Algebra is a powerful tool for analyzing
and designing logic circuits.
26Boolean Algebra
- Boolean algebra can be used to manipulate or
simplify Boolean expressions. - Why is this useful?
27Boolean Algebra
- Manipulating a Boolean expression results in an
alternate expression that is functionally
equivalent to the original. - Simplifying a Boolean expression results in an
expression with fewer logic operations and/or
fewer literals than the original. - The circuit diagram corresponding to the new
expression may be - Easier to build than the circuit diagram
corresponding to the original expression. - More cost effective than the circuit diagram
corresponding to the original expression.
28Basic Laws and Theorems
Operations with 0 and 1 1. X 0 X 1D. X 1
X 2. X 1 1 2D. X 0 0 Idempotent
laws 3. X X X 3D. X X X Involution
law 4. (X')' X Laws of complementarity 5. X
X' 1 5D. X X' 0
29Basic Laws and Theorems
Commutative laws 6. X Y Y X 6D. XY
YX Associative laws 7. (X Y) Z X (Y
Z) 7D. (XY)Z X(YZ) XYZ
X Y Z Distributive laws 8. X(Y Z)
XY XZ 8D. X YZ (X Y)(X Z)
Simplification theorems 9. XY XY' X
9D. (X Y)(X Y') X 10. X XY X 10D.
X(X Y) X 11. (X Y')Y XY 11D. XY' Y
X Y
30Basic Laws and Theorems
DeMorgan's laws 12. (X Y Z ...)'
X'Y'Z'... 12D. (XYZ...)' X' Y' Z' ...
Duality 13. (X Y Z ...)D XYZ... 13D.
(XYZ...)D X Y Z ... Theorem for
multiplying out and factoring 14. (X Y)(X'
Z) XZ X'Y 14D. XY X'Z (X Z)(X' Y)
Consensus theorem 15. XY YZ X'Z XY
X'Z 15D. (X Y)(Y Z)(X' Z) (X Y)(X' Z)
31Distributive Law Example 1
- Manipulate the following Boolean expression using
the distributive law - F (ABC').(A'BC)
- distributive law (8) X.(YZ) X.Y X.Z
32Distributive Law Example 2
- Manipulate the following Boolean expression using
the distributive law - F A.B'.C A'.B'.C'
- distributive law (8D) X Y.Z (XY).(XZ)
33Questions?