CSC Intro. to Computing Lecture 7: Circuits & Boolean Properties

Announcements Course slides are available from web page  Posted both before and after class  Slides after class includes my drawings Homework #2 handed out at end of class  Also available on course web page  Due by 5PM on Thursday, Feb. 9 CSC tutors are still available  Hours 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)

In-Class Exercise Diagram a)

In-Class Exercise Truth Table b)

In-Class Exercise Diagram b)

In-Class Exercise Truth Table c)

In-Class Exercise Diagram c)

In-Class Exercise Truth Table d)

In-Class Exercise Diagram d)

In-Class Exercise Truth Table e)

In-Class Exercise Diagram e)

Boolean Properties Law of Double Negation: PropertyANDOR Commutative a·b = b·aa+b = b+a Associative a·(b·c) = (a·b)·ca+(b+c)=(a+b)+c Distributive a·(b+c) = (a·b)+(a·c)a+(b·c) = (a+b)·(a+c) Identity a·1 = aa+0 = a Complement a·ā = 0a+ā = 1 DeMorgan a·b = ā+ba+b = ā·b Idempotency a·a = aa+a = a

More About Boolean Properties Properties identify equivalent circuits  E.g., circuits with identical truth table results Many ways we use these properties  Reduce delays by doing more work in parallel  Simplify circuits by removing useless gates

Using Boolean Properties CircuitProperty Used b+a Identity Reduce to a+b :

Using Boolean Properties CircuitProperty Used b+ab+a Identity a+b Reduce to a+b :

Using Boolean Properties CircuitProperty Used a·(a·b) Identity Reduce to a·b :

Using Boolean Properties CircuitProperty Used a·(a·b) Identity (a·a)·b Reduce to a·b :

Using Boolean Properties CircuitProperty Used a·(a·b) Identity (a·a)·b Associative a·b Reduce to a·b :

Using Boolean Properties CircuitProperty Used (c·d)+(d·c) Identity Reduce to c :

Using Boolean Properties CircuitProperty Used (c·d)+(d·c) Identity (c·d)+(c·d) Reduce to c :

Using Boolean Properties CircuitProperty Used (c·d)+(d·c) Identity (c·d)+(c·d) Commutative c·(d+d) Reduce to c :

Using Boolean Properties CircuitProperty Used (c·d)+(d·c) Identity (c·d)+(c·d) Commutative c·(d+d) Distributive c·1c·1 Reduce to c :

Using Boolean Properties CircuitProperty Used (c·d)+(d·c) Identity (c·d)+(c·d) Commutative c·(d+d) Distributive c·1 Complement c Reduce to c :

Using Boolean Properties CircuitProperty Used (y·z)·(z·y) Identity Reduce to y·z :

Using Boolean Properties CircuitProperty Used (y·z)·(z·y) Identity (y·z)·(y·z) Reduce to y·z :

Using Boolean Properties CircuitProperty Used (y·z)·(z·y) Identity (y·z)·(y·z) Commutative y·z Reduce to y·z :

DeMorgan’s Laws Two laws specific to logical systems  First stated in present form by Prof. DeMorgan  Useful for evaluating & simplifying circuits  Make great quiz questions, too

DeMorgan’s Laws Only properties that works with NAND or NOR gates  Easy to know when it should be used How to use DeMorgan’s Laws  Negate the inputs to the NAND/NOR gate  Replace the gate with its opposite NAND becomes an OR NOR becomes an AND

Using Boolean Properties CircuitProperty Used b·ab·a Identity Reduce to a+b :

Using Boolean Properties CircuitProperty Used b·ab·a Identity b+ab+a DeMorgan’s Reduce to a+b :

Using Boolean Properties CircuitProperty Used b·ab·a Identity b+ab+a DeMorgan’s b+ab+a Reduce to a+b :

Using Boolean Properties CircuitProperty Used b·ab·a Identity b+ab+a DeMorgan’s b+ab+a Double Negation a+ba+b Reduce to a+b :

Half-Adders Half-adder is a simple, but vital, circuit  Accepts two bits as input  Circuit then adds the two bits  Outputs 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 numbers  Takes two new bits and carry bit from last adder as inputs  Circuit adds all the bits  Outputs a result bit and another carry bit Full-adder is used to add additional bits  Also 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