Lecture Digital logic design - Lecture 4: Boolean algebra

The main contents of the chapter consist of the following: Basic logic functions can be made from AND, OR, and NOT (invert) functions; the behavior of digital circuits can be represented with waveforms, truth tables, or symbols; primitive gates can be combined to form larger circuits; boolean algebra defines how binary variables can be combined;…

Lecture 4

Boolean Algebra

• Today is Monday AND it is raining

• Today is Sunday OR it is NOT raining

• Today is Monday AND today is NOT Monday

- (This is a contradiction)

° The expression as a whole is either true or false.

Things can get a little tricky…

° Are these two statements equivalent?

• It is not nighttime and it is Monday OR it is raining and it is

Monday

• It is not nighttime or it is raining and Monday AND it is Monday

Boolean Algebra

° Formal logic: In formal logic, a statement

(proposition) is a declarative sentence that is either

Venn Diagrams

5

A  A B

B A

B A

B A

B A

B A

A

A A

Boolean Algebra

° Boolean Algebra is a mathematical technique that

provides the ability to algebraically simplify logic

expressions These simplified expressions will

result in a logic circuit that is equivalent to the

original circuit, yet requires fewer gates.

A

B

C

B+A C

A

B

Boolean Algebra, Logic and Gates

° Logical operators operate on binary values and binary

variables.

° Basic logical operators are the logic functions AND, OR and

NOT.

° Logic gates implement logic functions.

° Boolean Algebra: a useful mathematical system for

specifying and transforming logic functions.

° We study Boolean algebra as a foundation for designing

and analyzing digital systems!

° A literal is a Boolean variable or its complement A minterm

of the Boolean variables x 1 , x 2 , …, x n is a Boolean product

y 1 y 2 …y n , where y i = x i or y i = -x i

° Hence, a minterm is a product of n literals, with one literal

for each variable.

Boolean Switching

° Boolean (switching) variable x {0,1}∈

• 0, 1 are abstract symbols

They may correspond to {false, true} in logic, {off, on} of a switch, {low voltage, high voltage} of a CMOS circuit, or other meanings

° Logic functions with 1’s and 0’s

• Building digital circuitry

° We use 1 and 0 to denote the two values.

° Variable identifier examples:

• A, B, y, z, or X 1 for now

• RESET, START_IT, or ADD1 later

° NOT is denoted by an overbar ( ¯ ), a single quote

mark (') after, or (~) before the variable.

° Examples:

• is read “Y is equal to A AND B.”

• is read “z is equal to x OR y.”

• is read “X is equal to NOT A.”

Notation Examples

 Note: The statement:

1 + 1 = 2 (read “one plus one equals two”)

is not the same as

1 + 1 = 1 (read “1 or 1 equals 1”).

B A

Y

y x

z

A X

Operator Definitions

values "0" and "1" for each

0 1

1 0

X

NOT

X Z

Truth Tables

° Truth table a tabular listing of the values of a function

for all possible combinations of values on its arguments

° Example: Truth tables for the basic logic operations:

1 1 1

0 0 1

0 1 0

0 0 0

Z = X·Y Y

° Using Switches

• Inputs:

- logic 1 is switch closed

- logic 0 is switch open

• Outputs:

- logic 1 is light on

- logic 0 is light off.

• NOT input:

- logic 1 is switch open

- logic 0 is switch closed

° Example: Logic Using Switches

° Light is on (L = 1) for

L(A, B, C, D) = and off (L = 0), otherwise.

° Useful model for relay and CMOS gate

circuits, the foundation of current digital logic circuits

Logic Function Implementation – cont’d

BA

D

C

A (B C + D) = A B C + A D

Digital Systems

° Analysis problem:

• Determine binary outputs for each combination of inputs

° Design problem: given a task, develop a circuit

that accomplishes the task

• Many possible implementation

• Try to develop “best” circuit based on some criterion

(size, power, performance, etc.)

Logic Circuit

Toll Booth

Controller

° Consider the design of a toll booth controller

° Inputs: quarter, car sensor

° Outputs: gate lift signal, gate close signal

° If driver pitches in quarter, raise gate.

° When car has cleared gate, close gate.

Logic Circuit

$.25

Car?

Raise gate Close gate

Describing Circuit Functionality: Inverter

° Basic logic functions have symbols.

° The same functionality can be represented with truth

tables.

• Truth table completely specifies outputs for all input combinations.

° The above circuit is an inverter

° This is an AND gate

° So, if the two inputs signals

are asserted (high) the

output will also be asserted.

Otherwise, the output will

° So, if either of the two

input signals are

Describing Circuit Functionality: Waveforms

° Waveforms provide another approach for representing

functionality.

° Values are either high (logic 1) or low (logic 0).

° Can you create a truth table from the waveforms?

Consider three-input gates

3 Input OR Gate

Ordering Boolean Functions

Boolean Algebra

° A Boolean algebra is defined as a closed algebraic

system containing a set K or two or more elements and the two operators, and +.

° Useful for identifying and minimizing circuit

functionality

° Identity elements

• a + 0 = a

• a 1 = a

° 0 is the identity element for the + operation.

° 1 is the identity element for the operation.

Commutatively and Associativity of the Operators

Distributivity of the Operators and

° The Existence of the Complement:

For every a in K there exists a unique element

called a’ (complement of a) such that,

• a + a’ = 1

• a a’ = 0

° To simplify notation, the operator is frequently

omitted When two elements are written next to each other, the AND (.) operator is implied…

• a + b c = ( a + b ) ( a + c )

• a + bc = ( a + b )( a + c )

lity

° The principle of duality is an important concept

This says that if an expression is valid in Boolean algebra, the dual of that expression is also valid.

° To form the dual of an expression, replace all +

operators with operators, all operators with + operators, all ones with zeros, and all zeros with ones.

° Form the dual of the expression

a + (bc) = (a + b)(a + c)

° Following the replacement rules…

a(b + c) = ab + ac

° Take care not to alter the location of the

parentheses if they are present.

n

° This theorem states:

a’’ = a

° Remember that aa’ = 0 and a+a’=1

• Therefore, a’ is the complement of a and a is also the

complement of a’

• As the complement of a’ is unique, it follows that a’’=a.

° Taking the double inverse of a value will give the

initial value.

DeMorgan’s

Theorem

° A key theorem in simplifying Boolean algebra

expression is DeMorgan’s Theorem It states:

° Complement the expression

a(b + z(x + a’)) and simplify.

Y     X    Y X      14A)

Y X Y X X      13D)

Y X Y X X      13C)

Y X XY X

     13B)

Y X Y X X      13A)

YZ YW

XZ XW

Z W Y X      12B)

XZ XY

Z Y X      12A)

Z Y X Z

Y   X      11B)

Z XY YZ

X      11A)

X  

 Y 

 Y  X      10B)

X  

 Y 

 Y  X      10A)

Commutative  Law

Associative  Law

Distributiv

e Law

Consensus  Theorem

Boolean & DeMorgan’s Theorems

DeMorgan’s

A For Theorem #14B, break the line, and 

change the OR function to an AND function. 

Be sure to keep the lines over the variables.

° Basic logic functions can be made from AND, OR,

and NOT (invert) functions

° The behavior of digital circuits can be represented

with waveforms, truth tables, or symbols

° Primitive gates can be combined to form larger

circuits

° Boolean algebra defines how binary variables can

be combined

° Rules for associativity, commutativity, and

distribution are similar to algebra

° DeMorgan’s rules are important

• Will allow us to reduce circuit sizes.