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## ECE 331 – Digital System Design

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Truth Tables,

Boolean Expressions,

and

Boolean Algebra

(Lecture #3)

ECE 331 – Digital System DesignThe 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.

ECE 331 - Digital System DesignLogic 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.

ECE 331 - Digital System Design

Truth Tables

ECE 331 - Digital System DesignTruth 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.

4-input Truth Table

4 variables → 24 = 16 rows

ECE 331 - Digital System Design

F1 is completely defined by:

F2 is completely defined by:

Are F1 and F2 the same logic functions?

Truth Tables: ExamplesECE 331 - Digital System DesignBoolean 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

ECE 331 - Digital System DesignBoolean 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.

ECE 331 - Digital System Design

Using a truth table, evaluate the following Boolean expressions:

F1(A,B,C) = A'.B.C'

F2(A,B,C) = A + B' + C'

Boolean Expressions: Example #1Simple AND and OR Functions

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.

ECE 331 - Digital System Design

Using a truth table, evaluate the following Boolean expression:

F(A,B,C) = A'.C + B.C' + A.B'.C'

Boolean Expressions: Example #2More Complex Functions of ANDs and ORs

An AND term = 1

when all literals = 1.

An OR function = 1

when any term = 1.

ECE 331 - Digital System Design

Using a truth table, evaluate the following Boolean expression:

F(A,B,C) = (A+B').(A'+C).(A+B'+C')

Boolean Expressions: Example #3More Complex Functions of ANDs and ORs

An OR term = 1

when any literal = 1.

An AND function = 1

when all terms = 1.

B

logical operators

literals

F

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

ECE 331 - Digital System Design(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

ECE 331 - Digital System Design

Given the following truth table,

1. Derive a Boolean expression

2. Draw the corresponding circuit diagram

Logic Circuit: Example #1ECE 331 - Digital System Design

Given the following truth table,

1. Derive a Boolean expression

2. Draw the corresponding circuit diagram

Logic Circuit: Example #2ECE 331 - Digital System DesignEquivalency 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

ECE 331 - Digital System Design

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)'

Equivalence: ExampleECE 331 - Digital System Design

Boolean Algebra

ECE 331 - Digital System DesignBoolean 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.

ECE 331 - Digital System Design

Boolean algebra can be used to manipulate or simplify Boolean expressions.

Why is this useful?

Boolean AlgebraBoolean 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.

ECE 331 - Digital System DesignBasic 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

ECE 331 - Digital System DesignBasic 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

ECE 331 - Digital System DesignBasic 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)

ECE 331 - Digital System Design

Manipulate the following Boolean expression using the distributive law:

F = (A+B+C').(A'+B+C)

distributive law (8): X.(Y+Z) = X.Y + X.Z

Distributive Law: Example #1ECE 331 - Digital System Design

Manipulate the following Boolean expression using the distributive law:

F = A.B'.C + A'.B'.C'

distributive law (8D): X + Y.Z = (X+Y).(X+Z)

Distributive Law: Example #2ECE 331 - Digital System Design

Questions?

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