BOOLEAN LOGIC CSC 171 FALL 2004 LECTURE 7

ASSIGNMENT Review Quiz # 2 Start reading Chapter 5

History Boolean Logic 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

Quote “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 Idea Boolean Logic is two valued logic – True – False

JAVA The strings “true” and “false” are keywords in JAVA – Constants representing boolean value Variables can be declared to as boolean boolean b1, b2 ; Variables can be assigned a value b1 = true ; b2 = false ;

Relational Operators < // less than <= // less than or equal to > // greater than >= // greater than or equal to == // equal to != // not equal to

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

Example values ( 4 < 5) ( 4 <= 3) ( 3 > 3) ( 3 >= 3) (7 == 2) ( 2 != 7) true false true false true

Example assignments boolean 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 complex boolean b1, b2; b1 = ( 4 < (3 + 2)); int x = 7; b2 = ((x – 5) <= 3);

“AND” OPERATION ABA && B False TrueFalse TrueFalse True

Using AND boolean b1 ; double inc = ; int numChild = 5; 11 = ((inc 3));

“OR” OPERATION ABA || B False True FalseTrue

Using or boolean b2 ; double inc = ; int numChild = 5; b2 = ((inc 3));

“NOT” OPERATION A!A FALSETRUE FALSE

Using not boolean b3 ; double inc = ; int numChild = 5; b3 = !(!(inc 3));

Multiple modes of expression We can express booleans in many ways – Table – Formula – Symbolic diagram

The mathematical basis for computer 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 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 Arithmetic -(0) → 1 //looks funny -(1) → 0 // more like -1, → → → → 1 (no “carry”) 0 * 0 → 0 0 * 1 → 0 1 * 0 → 0 1 * 1 → 1 (intuitive)

Boolean Arithmetic -(0) → 1 // NOT -(1) → → → → → 1 OR 0 * 0 → 0 0 * 1 → 0 1 * 0 → 0 1 * 1 → 1 AND

Notation To keep things straight Boolean negation “!” replaces “-” Boolean addition “ || “ replaces “+” Boolean multiplication “&&” replaces “*”

Boolean Operations & Circuits AND OR NOT X

Boolean Expressions & Circuits AND: z = x && y ; OR: z = x || y ; NOT: z = !x ; X x y z x y z z x

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

So what? We can represent numbers in binary We can perform operations on boolean So, we can use boolean circuits to automate arithmetic operations

Binary Addition = = = = 10 A + B = CS

Addition = = = = 10 A + B = CS ABCS

QUESTION: “Carry in” = ?? = ?? = ?? = ?? = ?? = ?? = ?? = ??

QUESTION: “Carry in” = = = = = = = = 11