1. ECE 301 – Digital Electronics
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. ECE 301 - Digital Electronics
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.
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3. ECE 301 - Digital Electronics
Truth Tables
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4. ECE 301 - Digital Electronics
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.
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5. ECE 301 - Digital Electronics
3-input Truth Table # A B C
F(A,B,C) 1 2 3 4 5 6 7
3 variables → 23 = 8 rows
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6. 4-input Truth Table # A B C D F(A,B,C,D) 1 2 3 4 5 6 … 12 13 14 151 2 3 4 5 6 … 12 13 14 15
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4 variables → 24 = 16 rows

7. Truth Tables: Examples
F1 is completely defined by:
F2 is completely defined by:
Are F1 and F2 the same logic functions? A B C
F1(A,B,C) 1 A B C
F2(A,B,C) 1
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8. ECE 301 - Digital Electronics
Boolean Expressions
(and Logic Circuits)
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9. ECE 301 - Digital Electronics
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'+B+C).(A+B'+C).(A+B+C')
F3 = A'.(B+C) + B.(A+C')
Literals highlighted in green
Logical operators highlighted in blue
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10. ECE 301 - Digital Electronics
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.
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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.
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12. Boolean Expressions: Example #1C A' B' C' F1 F2 1
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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.
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14. Boolean Expressions: Example #2C
A'C
BC'
AB'C'
F(A,B,C) 1
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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) = (A+B').(A'+C).(A+B'+C')
An OR term = 1
when any literal = 1.
An AND function = 1
when all terms = 1.
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16. Boolean Expressions: Example #3C
A+B'
A'+C
A+B'+C'
F(A,B,C) 1
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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
logical operators
literals A B F
Circuit Diagram
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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
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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) 1
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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) 1
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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
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22. ECE 301 - Digital Electronics
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)'
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23. ECE 301 - Digital Electronics
Equivalence: Example A B
A'B
AB'
A'B' AB F1 F2 1
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24. ECE 301 - Digital Electronics
Boolean Algebra
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25. ECE 301 - Digital Electronics
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.
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ECE Digital Electronics

26. ECE 301 - Digital Electronics
Boolean Algebra
Boolean algebra can be used to manipulate or simplify Boolean expressions.
Why is this useful?
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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.
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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
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29. Basic Laws and Theorems
Commutative laws:
6. X + Y = Y + X D. 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 D. (X + Y)(X + Y') = X
10. X + XY = X D. X(X + Y) = X
11. (X + Y')Y = XY D. XY' + Y = X + Y
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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 D. (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)
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31. Distributive Law: Example #1
Distributive law (8): X.(Y + Z) = X.Y + X.Z
Use the distributive law to multiply out the following Boolean expression:
F = (AB+C).(D'+E)
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32. Distributive Law: Example #2
Distributive law (8D): X + Y.Z = (X+Y).(X+Z)
Use the distributive law to factor the following Boolean expression:
F = A'.B + C.D'.E
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33. ECE 301 - Digital Electronics
Questions?
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ECE Digital Electronics