Logic Gates II Informatics INFO I101 February 5, 2003 John C. Paolillo, Instructor

Items for Today Last time –Logic Circuits –Applications of Boolean Logic This time –More Circuits –Addition and subtraction –Two’s complement representation –Color

Quiz Answers and (1) or (2) C A B Exclusive OR (XOR) xor (3)

Quiz: 10 minutes Take out a sheet of paper Write you name on it and date it Answer the following questions (next slide)

Quiz Questions (1)Give the truth table for Boolean AND (2)Give the truth table for Boolean OR (3)What does the following circuit do? Give a truth table for it, and name the logic gate it corresponds to. C A B

Graphic Paint/Copy Modes COPYORXOR

Arithmetic Addition and Subtraction

Addition: Half Adder S A B xor and C The half adder sends a carry, but can’t accept one +

Addition: Full Adder A B S CO CI

Addition: Truth Tables CI 0 1 A A B B SCO S S

More Digits Full Adder ci co s abab a0 s0 b0 Full Adder ci co s abab a1 s1 b1 Full Adder ci co s abab a2 s2 b2 Full Adder ci co s abab a3 s3 b3 Full adders can be cascaded

Subtraction – –0100 Subtraction is asymmetrical That makes it harder We have to borrow sometimes

Solution: “Easy Subtraction” 456 – – Subtraction is easy if you don’t have to borrow i.e. if all the digits of the minuend are greater than (or equal to) all those of the subtrahend This will always be true if the minuend is all 9’s: 999, or , or etc.

How can we use easy subtraction? Subtract the subtrahend from 999 (or whatever we need) (easy) Add the result to the minuend (easy enough) Add 1 (easy) Subtract 1000 (not too hard) Difference = Minuend – Subtrahend + 1 – 1000 This works for binary as well as decimal

Subtraction Example – ????????? – This is the same as inverting each bit – Regular addition Add one Now drop the highest bit (easy: it’s out of range)

Subtraction Procedure Invert each bit Regular addition Add one Now drop the highest bit (easy: it’s out of range) Each of these steps is a simple operation we can perform using our logic circuits Bitwise XOR Cascaded Adders Add carry bit Drop the highest bit (overflows)

Negative Numbers Invert each bit Add one These steps make the negative of a number in twos-complement notation Twos complements can be added to other numbers normally Positive numbers cannot use the highest bit (the sign bit) This is the normal representation of negative numbers in binary

Counting etc – – – – – – – – – –10 etc.

Representations The number representation you use (encoding) affects the way you need to do arithmetic (procedure) This is true of all codes: encoding (representation) affects procedure (algorithm) Good binary codes make use of properties of binary numbers and digital logic

Color (review)

The Eight-Color Computer Eight colors: black, yellow, magenta, red, cyan, green, blue, white Three color tubes on a TV monitor: Red, Green, Blue 2 3 =8 Additive color relations: red+green+blue=white

Color Perception 3 Electron guns, aimed at 3 different colors of phosphor dots 3 types of retinal sensor cells, sensitive to 3 different bands of light

Color: Response Patterns red cones green cones blue cones Wavelength 

Color: Neural Encoding Wavelength  110 RGB