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

Announcements Quiz at end of class on Thursday  Covers 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 good  Most mistakes came from miscopying numbers or addition mistakes  Very few mistakes were due to errors in algorithm Brief review includes most common error(s)

Homework Problem #7 Convert 14D 16 into octal  Don’t know how to convert hex to octal directly  (Harder) Solution: Convert to decimal first 14D 16 =

Convert 14D 16 to Octal (Harder) Solution: Convert to decimal first 14D 16 = 13 * 16 0 = 13 * 1 = 13 4 * 16 1 = 4 * 16 = 64 1 * 16 2 = 1 * 256 =

Convert 14D 16 to Octal (Harder) Solution: Convert to decimal first 14D 16 = = 515 8

Convert to Hexadecimal (Easier) Method: Convert to binary first =

Convert to Hexadecimal (Easier) Method: Convert to binary first = Convert 5 Convert 1

Convert to Hexadecimal (Easier) Method: Convert to binary first = 2 = = 1 * 2 0 = 1 * 1 = 1 0 * 2 1 = 0 * 2 = 0 1 * 2 2 = 1 * 4 = 4 1 * 2 3 = 1 * 8 =

Convert to Hexadecimal (Easier) Method: Convert to binary first = 2 = = 1 * 2 0 = 1 * 1 = 1 0 * 2 1 = 0 * 2 = 0 1 * 2 2 = 1 * 4 = 4 1 * 2 3 = 1 * 8 = = D D

Convert to Hexadecimal (Easier) Method: Convert to binary first = 2 = = 0 * 2 0 = 0 * 1 = 0 0 * 2 1 = 0 * 2 = 0 1 * 2 2 = 1 * 4 = 4 0 * 2 3 = 0 * 8 = = D

Convert to Hexadecimal (Easier) Method: Convert to binary first = 2 = = 1 * 2 0 = 1 * 1 = = D

George Boole Mathematician from English middle-class  Lived from 1815 – 1864  Started 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 system  World 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 operations  You have been using some of these for years, however  Work 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 operation  Returns the negation of the input True when a is false; false when a is true  Written as false true

Boolean Operation: OR OR takes two values as inputs  Written as  Returns true when either value is true True when either a OR b are true; false otherwise false true false true

Boolean Operation: AND AND also takes two inputs  Written as  Returns true when both values are true True when a AND b are true; false otherwise false true false true

Boolean Operation: XOR First slightly unfamiliar idea: XOR  XOR computes “exclusive or”  Written as  True when a or b, but not both, is true false true false true

Boolean Operation: NAND NAND computes NEGATIVE AND  Written as Combines notation for AND & NOT  Returns true when either input is not true True if a OR b are false; false otherwise false true false true

Boolean Operation: NOR Last operation is NEGATIVE OR  Written 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 false true false true

George Boole: 1World: 0 Turns out Boolean algebra is VITAL  Basis 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 terms  For more on truth, see religious studies or philosophy Use “1” rather than true & “0” for false

Extended Boole’s Algebra Algebra includes ability to combine inputs and operations functions  f(x, y, z) = x 2 + y + z  f(2, 4, 1)= = = 14 This is true for Boole’s algebra, too  Computers do more than negate bits  We 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 0 1

Even More Complex Circuits Compute  Negate a, then compute result AND b

Really Complex Circuits Determine truth table for

More Complex Circuits Consider

Creating Truth Tables Start truth table by listing all possible inputs  For 1 input, 2 possible values exist  For 2 inputs, 4 possible combinations  For 3 inputs, 8 combinations  For 4 inputs, 16 combinations  For 5 inputs, 32 combinations  For 10 inputs, 1024 combinations  For n inputs, 2 n combinations

Creating Truth Tables There is a secret to making the inputs  1 st input – Alternate one 0 and one 1  2 nd input – Repeat two 0s then two 1s  3 rd input – Repeat four 0s then four 1s  4 th 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

Gate Piece of hardware which combines electrical signal input(s) to generate an output signal  Computer 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 operations  Similar 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 b a b a b

Drawing Logic Diagrams Often use results from one gate as input into another gate  E.g., use result from in Show this by connecting wire from first gate’s output to second gate’s input b c a

a b c Drawing Logic Diagrams Sometimes we have one value used as input for more than 1 gate  Show this by drawing a “splitter”  E.g.

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