cec 220 digital circuit design boolean algebra i wed, sept 2 cec 220 digital circuit design slide 1...

CEC 220 Digital Circuit DesignBoolean Algebra I

Wed, Sept 2 CEC 220 Digital Circuit Design Slide 1 of 13

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Lecture Outline

Wed, Sept 2 CEC 220 Digital Circuit Design

• Introduction• Basic Operations: NOT, AND, OR• Representations of Boolean Expressions

Slide 2 of 13


Boolean AlgebraIntroduction


• Boolean Algebra• In 1849 George Boole published a scheme for the algebraic

description of logic processes

• In the 1930’s Claude Shannon used Boolean Algebra to describe circuits built with switches

• Boolean Algebra is an effective tool for describing logic circuits

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Boolean AlgebraBoolean Logic


• Two logic levels TRUE = HIGH = 1 FALSE = LOW = 0

+5

V

0

1 0 1

T ime

Transition from logic 1 to logic 0does not take place instantaneouslyin real digital systems

+5

V

0

Logic 1

Logic 0 Intermediate values may be visiblefor an instant

Time

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Boolean AlgebraBasic Operations: NOT, AND, OR


• Logical NOT Description:

o The output is the complement/inverse/opposite of the input

Symbolic Representation (NOT gate):

Truth Table Representation:

Boolean Description: C = NOT A

A C

A C0 11 0

A CF TT F

or C = A’ or C =

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• Logical AND Description:

o The output is TRUE if and only if all the inputs are TRUE

Symbolic Representation (AND gate):


Boolean Description: C = A AND B

A B C0 0 00 1 01 0 01 1 1

A B CF F FF T FT F FT T T

or C = AB or C = A B

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• Logical OR Description:

o The output is TRUE if any of the inputs are TRUE

Symbolic Representation (OR gate):


Boolean Description: C = A OR B

A B C0 0 00 1 11 0 11 1 1

A B CF F FF T TT F TT T T

or C = A + B

This is NOT addition!!Slide 7 of 13


Boolean AlgebraLogic Gates and Boolean Expressions


• Derive an expression for the output of this logic circuit? Eventually we will omit the “” in the AND gate and “+” in

the OR gate

• The logic expression is a function of three variables (A, B, and C).

B’

AB’AB’+C

Higher precedence!!

Lower precedence!!

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• Derive an expression for the output of this logic circuit?

• The output is which may also be written as [A(C+D)]’+BE This expression has five variables (A, B, C, D, and E)

C+D A(C+D) A(C+D )

BE

+

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• Literals Each appearance of a variable or its complement in an

expression is referred to as a literal. Example:

The expression has three variables (A, B, and C) The expression has 10 literals

AB’C+A’B+A’BC’+B’C’

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Boolean AlgebraTruth Tables of a Logic Circuit


• Determine the truth table for the output (F) of the logic circuit

A B0 00 11 01 1

A’1100

F=A’+B1101

Two variablesFour possible inputs (i.e. 2n)

Truth table lists ALL possible inputs and associated outputs

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A+C01011111

Boolean AlgebraTruth Tables of a Logic Circuit


• Determine the truth table for the output of the logic circuit A

C

BB’

B’+C

A+C

(A+C)(B’+C)

B’+C11011101

A+C B’+C (A+C)(B’+C)01011101

(A+C)(B’+C)B’11001100

A00001111

B00110011

C01010101

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Next Lecture


• Basic Boolean Theorems• Implementation of Boolean Expressions

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cec 220 digital circuit design boolean algebra i wed, sept 2 cec 220 digital circuit design slide 1...

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