CSC 110 - Intro. to Computing Lecture 5: Boolean Logic, Gates, & Circuits

CSC 110 -Intro. to Computing

Lecture 5:

Boolean Logic, Gates, & Circuits

Announcements

Quiz at end of class on ThursdayCovers converting numbers between bases

and addition in other bases Homework answers available on web

Homework #2 will be next week CSC tutors are now available

Hours posted outside Wehle 206 & 208

Homework Review

Results were (generally) very goodMost mistakes came from miscopying

numbers or addition mistakesVery few mistakes were due to errors in

algorithm Brief review includes most common

error(s)

Homework Problem #7

Convert 14D16 into octalDon’t know how to convert hex to octal

directly(Harder) Solution: Convert to decimal first

14D16 =

Convert 14D16 to Octal

(Harder) Solution: Convert to decimal first14D16 = 13 * 160 = 13 * 1 = 13

4 * 161 = 4 * 16 = 64 1 * 162 = 1 * 256 = + 256

33310

Convert 14D16 to Octal

(Harder) Solution: Convert to decimal first14D16 = 33310 = 5158

41 5 08 333 8 41 8 5

32 40 0 13 1 5 8 5

Convert 5158 to Hexadecimal

(Easier) Method: Convert to binary first5158 =


(Easier) Method: Convert to binary first5158 = 101 001 1012

Convert 5

Convert 1

2 1 02 5 2 2 2 1

4 2 01 0 1

02 1 Need to pad wi th 2 0s to make 3 bi ts!

01


(Easier) Method: Convert to binary first5158 = 2 = 16

11012 = 1 * 20 = 1 * 1 = 1 0 * 21 = 0 * 2 = 0 1 * 22 = 1 * 4 = 4 1 * 23 = 1 * 8 = + 8 1310

10100

1101



11012 = 1 * 20 = 1 * 1 = 1 0 * 21 = 0 * 2 = 0 1 * 22 = 1 * 4 = 4 1 * 23 = 1 * 8 = + 8 1310 = D16

10100

1101

D



01002 = 0 * 20 = 0 * 1 = 0 0 * 21 = 0 * 2 = 0 1 * 22 = 1 * 4 = 4 0 * 23 = 0 * 8 = + 0 410 = 416

10100

1101

4 D



12 = 1 * 20 = 1 * 1 = 1 110 = 116

10100

1101

1 4 D

George Boole

Mathematician from English middle-classLived from 1815 – 1864Started work at age 16 as a teaching assistant

Held two assistantships to support family Later opened his own school Reached pinnacle when appointed a Professor

Published Mathematical Analysis of Logic in 1847

Mathematical Analysis of Logic

Boole’s book proposed new logical systemWorld consisted of only two values

Derived other numbers from combinations of these

Devised addition, subtraction, and multiplication rules

Basis for several areas of academic research World thought work was of little importance

Not until 1963 did someone find a use for it…

Boolean System of Logic

Also called “Boolean algebra” Initial world consists of two values

Think of them as “true” and “false” Also defines several basic Boolean

operationsYou have been using some of these for years,

howeverWork exactly as you have been using them

Truth Table

Normal way we present Boolean functions Each table include all possible inputs…

Made easier since we only have true & false …and shows each possible result Not as complicated as it sounds

Actually, not very complicated at all!

Boolean Negation: NOT

NOT is simplest operationReturns the negation of the input

True when a is false; false when a is true

Written as a

false

true

aa

Boolean Operation: OR

OR takes two values as inputsWritten as Returns true when either value is true

True when either a OR b are true; false otherwise

false falsefalse truetrue falsetrue true

a b a b

a b

Boolean Operation: AND

AND also takes two inputsWritten asReturns true when both values are true

True when a AND b are true; false otherwise


a b a b

a b

Boolean Operation: XOR

First slightly unfamiliar idea: XORXOR computes “exclusive or” Written as True when a or b, but not both, is true


a b a b

a b

Boolean Operation: NAND

NAND computes NEGATIVE ANDWritten as

Combines notation for AND & NOT

Returns true when either input is not true True if a OR b are false; false otherwise


a b a b

a b

Boolean Operation: NOR

Last operation is NEGATIVE ORWritten as

The negation (NOT) of an OR operation

Returns true when both inputs are not true True if a AND b are false; false otherwise


a b a b

a b

George Boole: 1 World: 0

Turns out Boolean algebra is VITALBasis for nearly all computer hardware work

“Last useful thing by anyone claimed by computer scientists”– a former colleague (hardware engineer)

Boolean Logic in Computers

Computers work best with concrete termsFor more on truth, see religious studies or

philosophy Use “1” rather than true & “0” for false

0 00 11 01 1

a b a b

Extended Boole’s Algebra

Algebra includes ability to combine inputs and operations functions f(x, y, z) = x2 + y + z3 + 5 f(2, 4, 1) = 22 + 4 + 13 + 5

= 4 + 4 + 1 + 5 = 14

This is true for Boole’s algebra, tooComputers do more than negate bitsWe still use truth tables for this work

Circuits

Combines results of multiple operations List result of each operation in truth table

Compute each new operation step-by-step Consider combining two NOT operations

Written a

0

1

aa a

Even More Complex Circuits

Compute Negate a, then compute result AND b

a b

0 00 11 01 1

a b a a b

Really Complex Circuits

Determine truth table for ( ) ( )a b a b

0 00 11 01 1

a b a b a b ( ) ( )a b a b

More Complex Circuits

Consider ( )a b c

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

a b c

Creating Truth Tables

Start truth table by listing all possible inputsFor 1 input, 2 possible values existFor 2 inputs, 4 possible combinationsFor 3 inputs, 8 combinationsFor 4 inputs, 16 combinations For 5 inputs, 32 combinationsFor 10 inputs, 1024 combinationsFor n inputs, 2n combinations

Creating Truth Tables

There is a secret to making the inputs1st input – Alternate one 0 and one 12nd input – Repeat two 0s then two 1s3rd input – Repeat four 0s then four 1s4th input – Repeat eight 0s then eight 1s

Start with row of all 0s Stop when you reach a row with all 1s

Creating Truth Table

Keep going until we reach row with only 1s

0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1

a b c0 00 11 01 1

a b01

a

Gate

Piece of hardware which combines electrical signal input(s) to generate an output signalComputer signals range from 0 – 5 volts

0 – 2 volts is “low” --- in “0” state 2 – 5 volts is “high” and in “1” state

6 gates commonly used today:NOT, OR, AND, XOR, NAND, NOR

Logic Diagrams

Gate-centric way of writing Boolean operationsSimilar to equations from before, but not as

math-centric Highlights the electronic nature of gates

Shows values coming into and out of each gate as a wire

Label the wires for circuit inputs

Logic Diagrams

a

a

b

a

b

a

a b

a b

Logic Diagramsa

b

a

b

a

b

a b

a b

a b

Drawing Logic Diagrams

Often use results from one gate as input into another gateE.g., use result from in

Show this by connecting wire from first gate’s output to second gate’s input

( ) a b cb c

b

c

a

a

b

c

Drawing Logic Diagrams

Sometimes we have one value used as input for more than 1 gateShow this by drawing a “splitter”E.g. ( ) ( ) a b b c

For next lecture

Study for the quiz on number systems on Thursday

Finish reading section 4 Be ready to discuss

Algebraic Properties of Boolean Logic

Documents

CSC 110 - Intro. to Computing Lecture 5: Boolean Logic, Gates, & Circuits