CSC 110 -Intro. to Computing

Lecture 7:

Circuits & Boolean Properties

Announcements

Course slides are available from web pagePosted both before and after classSlides after class includes my drawings

Homework #2 handed out at end of classAlso available on course web pageDue by 5PM on Thursday, Feb. 9

CSC tutors are still availableHours posted outside Wehle 206 & 208

Announcements

Quiz #1 handed back at end of class Mean score: 82 Standard deviation: 20 Answer key available on web page

Lowest quiz and homework score is dropped I expect everyone will still get a 100% for this course

Do not worry about your difficulties in mathematics. I can assure you mine are still greater.

-- Albert Einstein

In-Class Exercise Truth Table

a) a b

In-Class Exercise Diagram

a) a b

In-Class Exercise Truth Table

b) a b

In-Class Exercise Diagram

b) a b

In-Class Exercise Truth Table

c) ( ) a b a

In-Class Exercise Diagram

c) ( ) a b a

In-Class Exercise Truth Table

d) ( ) c a b

In-Class Exercise Diagram

d) ( ) c a b

In-Class Exercise Truth Table

e) ( ) ( ) c a a b

In-Class Exercise Diagram

e) ( ) ( ) c a a b

Boolean Properties

Law of Double Negation: a a

Property AND OR

Commutative a·b = b·a a+b = b+a

Associative a·(b·c) = (a·b)·c a+(b+c)=(a+b)+c

Distributive a·(b+c) = (a·b)+(a·c) a+(b·c) = (a+b)·(a+c)

Identity a·1 = a a+0 = a

Complement a·ā = 0 a+ā = 1

DeMorgan a·b = ā+b a+b = ā·b

Idempotency a·a = a a+a = a

More About Boolean Properties

Properties identify equivalent circuitsE.g., circuits with identical truth table results

Many ways we use these propertiesReduce delays by doing more work in parallelSimplify circuits by removing useless gates

0 1 0

1 0 1

aa a

Using Boolean Properties

Circuit Property Usedb+a Identity

Reduce to a+b:

Using Boolean Properties

Circuit Property Usedb+a Identitya+b

Reduce to a+b:

Using Boolean Properties

Circuit Property Useda·(a·b) Identity

Reduce to a·b:

Using Boolean Properties

Circuit Property Useda·(a·b) Identity(a·a)·b

Reduce to a·b:

Using Boolean Properties

Circuit Property Useda·(a·b) Identity(a·a)·b Associative

a·b

Reduce to a·b:

Using Boolean Properties

Circuit Property Used(c·d)+(d·c) Identity

Reduce to c:

Using Boolean Properties

Circuit Property Used(c·d)+(d·c) Identity(c·d)+(c·d)

Reduce to c:

Using Boolean Properties

Circuit Property Used(c·d)+(d·c) Identity(c·d)+(c·d) Commutative

c·(d+d)

Reduce to c:

Using Boolean Properties

Circuit Property Used(c·d)+(d·c) Identity(c·d)+(c·d) Commutative

c·(d+d) Distributivec·1

Reduce to c:

Using Boolean Properties

Circuit Property Used(c·d)+(d·c) Identity(c·d)+(c·d) Commutative

c·(d+d) Distributivec·1 Complementc

Reduce to c:

Using Boolean Properties

Circuit Property Used(y·z)·(z·y) Identity

Reduce to y·z:

Using Boolean Properties

Circuit Property Used(y·z)·(z·y) Identity(y·z)·(y·z)

Reduce to y·z:

Using Boolean Properties

Circuit Property Used(y·z)·(z·y) Identity(y·z)·(y·z) Commutative

y·z

Reduce to y·z:

DeMorgan’s Laws

Two laws specific to logicalsystemsFirst stated in present form by Prof.

DeMorganUseful for evaluating & simplifying

circuitsMake great quiz questions, too

DeMorgan’s Laws

Only properties that works with NAND or NOR gatesEasy to know when it should be used

How to use DeMorgan’s LawsNegate the inputs to the NAND/NOR gateReplace the gate with its opposite

NAND becomes an OR NOR becomes an AND

Using Boolean Properties

Circuit Property Usedb·a Identity

Reduce to a+b:

Using Boolean Properties

Circuit Property Usedb·a Identityb+a DeMorgan’s

Reduce to a+b:

Using Boolean Properties

Circuit Property Usedb·a Identityb+a DeMorgan’sb+a

Reduce to a+b:

Using Boolean Properties

Circuit Property Usedb·a Identityb+a DeMorgan’sb+a Double Negationa+b

Reduce to a+b:

Half-Adders

Half-adder is a simple, but vital, circuitAccepts two bits as inputCircuit then adds the two bitsOutputs the result bit and a carry bit

Half-adder can only be used to add least significant bit of a large number

Half-Adder

Which line is result and which is carry?

Full-Adders

Full-adder continues adding two numbersTakes two new bits and carry bit from last

adder as inputsCircuit adds all the bitsOutputs a result bit and another carry bit

Full-adder is used to add additional bitsAlso makes EXCELLENT quiz and midterm

questions

Full-Adder

Slightly more complex version of a half-adder

For next lecture

Do your homework! Start reading Section 5 Be ready to discuss:

Computer components