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ECE 331

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Title: ECE 331


1
ECE 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.
2
Logic 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
4
Truth 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.

5
3-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
6
4-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
7
Truth 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)
9
Boolean 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
10
Boolean 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.

11
Boolean 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.
12
Boolean 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
13
Boolean 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.
14
Boolean 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
15
Boolean 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.
16
Boolean 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
17
Logic 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

19
Logic 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
20
Logic 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
21
Equivalency 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

22
Equivalence 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)'

23
Equivalence Example
A B A'B AB' A'B' AB F1 F2
0 0
0 1
1 0
1 1
24
Boolean Algebra
25
Boolean 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.

26
Boolean Algebra
  • Boolean algebra can be used to manipulate or
    simplify Boolean expressions.
  • Why is this useful?

27
Boolean 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.

28
Basic 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  
29
Basic 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
30
Basic 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)
31
Distributive 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

32
Distributive 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)

33
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