BOOLEAN LOGICBOOLEAN LOGIC

CSC 171 FALL 2004

LECTURE 7

ASSIGNMENTASSIGNMENT

Review Quiz # 2

Start reading Chapter 5

HistoryHistoryBoolean LogicBoolean Logic

1854 - George Boole describes his system for symbolic and logical reasoning:

An Investigation into the Laws of thought

Boolean logic later becomes the basis for computer design

QuoteQuote

“No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.”

George Boole - Quoted in D MacHale, Comic Sections (Dublin 1993)

Basic IdeaBasic Idea

Boolean Logic is two valued logic– True– False

JAVAJAVA

The strings “true” and “false” are keywords in JAVA– Constants representing boolean value

Variables can be declared to as booleanboolean b1, b2 ;

Variables can be assigned a valueb1 = true ;b2 = false ;

Relational OperatorsRelational Operators< // less than<= // less than or equal to> // greater than>= // greater than or equal

to == // equal to!= // not equal to

ExamplesExamples( 4 < 5) ( 4 <= 3) ( 3 > 3)( 3 >= 3) (7 == 2)( 2 != 7)

Example valuesExample values( 4 < 5)

( 4 <= 3) ( 3 > 3)( 3 >= 3) (7 == 2)( 2 != 7)

truefalsefalsetruefalsetrue

Example assignmentsExample assignmentsboolean b1, b2, b3, b4, b5, b6;b1 = ( 4 < 5); b2 = ( 4 <= 3); b3 = ( 3 > 3);b4 = ( 3 >= 3); b5 = (7 == 2);b6 = ( 2 != 7);

Can be more complexCan be more complexboolean b1, b2;b1 = ( 4 < (3 + 2));int x = 7; b2 = ((x – 5) <= 3);

““AND” OPERATIONAND” OPERATION

A B A && B

False False False

False True False

True False False

True True True

Using ANDUsing ANDboolean b1 ;double inc = 73000.0;int numChild = 5;11 = ((inc < 50000.0) && (numChild >

3));

““OR” OPERATIONOR” OPERATION

A B A || B

False False False

False True True

True False True

True True True

Using orUsing orboolean b2 ;double inc = 73000.0;int numChild = 5;b2 = ((inc < 50000.0) || (numChild >

3));

““NOT” OPERATIONNOT” OPERATION

A !A

FALSE TRUE

TRUE FALSE

Using notUsing notboolean b3 ;double inc = 73000.0;int numChild = 5;b3 = !(!(inc < 50000.0) && !(numChild >

3));

Multiple modes of expressionMultiple modes of expression

We can express booleans in many ways– Table– Formula– Symbolic diagram

The mathematical basis for The mathematical basis for computer designcomputer design

An open switch can represent– Zero– False

A closed switch can represent– One– True

A switch can control another switch– A logic gate

Boolean ArithmeticBoolean Arithmetic

-True → False

-False → True

False + False → False

False + True → True

True + False → True

True + True → True

False * False → False

False * True → False

True * False → False

True * True → True

Boolean ArithmeticBoolean Arithmetic

-(0) → 1 //looks funny

-(1) → 0 // more like -1,1

0 + 0 → 0

0 + 1 → 1

1 + 0 → 1

1 + 1 → 1

(no “carry”)

0 * 0 → 0

0 * 1 → 0

1 * 0 → 0

1 * 1 → 1

(intuitive)

Boolean ArithmeticBoolean Arithmetic

-(0) → 1 // NOT

-(1) → 0

0 + 0 → 0

0 + 1 → 1

1 + 0 → 1

1 + 1 → 1

OR

0 * 0 → 0

0 * 1 → 0

1 * 0 → 0

1 * 1 → 1

AND

NotationNotation

To keep things straight

Boolean negation “!” replaces “-”Boolean addition “ || “ replaces “+”Boolean multiplication “&&” replaces “*”

Boolean Operations & CircuitsBoolean Operations & Circuits

AND

OR

NOT

X

Boolean Expressions & CircuitsBoolean Expressions & Circuits

AND: z = x && y ;

OR: z = x || y ;

NOT: z = !x ;

X

xy z

xy z

zx

Some practiceSome practice

Express the “circuit” as– A truth table– A formula (boolean assignment in JAVA)

So what?So what?

We can represent numbers in binaryWe can perform operations on booleanSo, we can use boolean circuits to automate

arithmetic operations

Binary AdditionBinary Addition

0 + 0 = 00

0 + 1 = 01

1 + 0 = 01

1 + 1 = 10

A + B = CS

AdditionAddition

0 + 0 = 00

0 + 1 = 01

1 + 0 = 01

1 + 1 = 10

A + B = CS

A B C S

0 0 0 0

0 1 0 1

1 0 0 1

1 1 0 0

QUESTION: “Carry in”QUESTION: “Carry in”

0 + 0 + 0 = ??

0 + 0 + 1 = ??

0 + 1 + 0 = ??

0 + 1 + 1 = ??

1 + 0 + 0 = ??

1 + 0 + 1 = ??

1 + 1 + 0 = ??

1 + 1 + 1 = ??

QUESTION: “Carry in”QUESTION: “Carry in”

0 + 0 + 0 = 00

0 + 0 + 1 = 01

0 + 1 + 0 = 01

0 + 1 + 1 = 10

1 + 0 + 0 = 01

1 + 0 + 1 = 10

1 + 1 + 0 = 10

1 + 1 + 1 = 11