# BOOlean Logic The basis for digital electronics..

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BOOlean Logic The basis for digital electronics.

What digital electronics do you use?  Computer  CD & DVD players  IPod  Cell phone  HDTV  Digital cameras

What are digital electronics?  Sound is an analog signal.  On a CD, digital sound is encoded as 44.1 kHz, 16 bit audio.  The original wave is 'sliced' 44,100 times a second - and an average amplitude level is applied to each sample.  16 bit means that a total of 65,536 different values can be assigned, or quantized to each sample.  DVD-Audio can be 96 or 192 kHz and up to 24 bits resolution

George Boole  George Boole, a British mathematician (1815-1864).  Logic and math are equivalent.  All math functions can be determined using these 3 primary Boolean logic operators: AND, OR, and NOT.  AND narrows your search,  OR broadens your search, and  NOT is used to exclude concepts.

Have you used Boolean logic?  What information have you searched for on the Web?  Do a Google Search for car radios.

The AND operator (both, all)  rivers AND salinity  dairy products AND export AND Europe

The OR operator (either, any)  fruit OR vegetables  fruit OR vegetables OR cereal from CSIRO Australia

The NOT operator  fruit NOT apples  Look at the Boolean Machine http://kathyschrock.net/rbs3k/boolean/

Let’s use Boolean logic to examine class.  Please stand up if you are:  girl  AND black hair  AND left handed  Please stand up if you are:  girl  OR black hair  OR left handed  And NOT  How has the group changed depending on the logical operator used.

What would a library search give for “stress” and “anxiety”? Shaded areas indicate retrieval Search StatementTypes of records retrieved Number of records retrieved stress OR anxiety Documents that discuss either stress or anxiety 1127 stress AND anxiety Documents that discuss both stress and anxiety 51 stress NOT anxiety Documents that discuss stress but not anxiety 687

Nesting  When more than one element is in parentheses, the sequence is left to right. This is called "nesting."  (foxes OR rabbits) AND pest control  foxes OR rabbits AND pest control  (animal pests OR pest animals) NOT rabbits  Parentheses should be used to group terms joined by OR when there is any other operator in the search.

Order of precedence of Boolean operators  The order of operations is: AND, NOT, OR, XOR  Parentheses are used to override priority.  Expressions in parentheses are processed first.  Parentheses are used to organize the sequence and groups of concepts.

Write out logic statements using Boolean operators for these.  You have a buzzer in your car that sounds when your keys are in the ignition and the door is open.  You have a fire alarm installed in your house. This alarm will sound if it senses heat or smoke.  There is an election coming up. People are allowed to vote if they are a citizen and they are 18.  To complete an assignment the students must do a presentation or write an essay.

Basis for digital computers.  The true-false nature of Boolean logic makes it compatible with binary logic used in digital computers.  Electronic circuits can produce Boolean logic operations.  Circuits are called gates.  NOT  AND  OR

NOT gate  The simplest possible gate is called an "inverter," or a NOT gate.  One bit as input produces its opposite as output.  The symbol for a NOT gate in circuit diagrams is shown below.  The logic table for the NOT gate shows input and output. AQ 01 10 From How Stuff Works

AND gate  The AND gate has the following symbol and logic table.  Two or more input bits produce one output bit.  Both inputs must be true (1) for the output to be true.  Otherwise the output is false (0). ABQ 000 010 100 111

OR gate  The OR gate has the following symbol and logic table.  Two or more input bits produce one output bit.  Either inputs must be true (1) for the output to be true.  http://www.play-hookey.com/digital/basic_gates.html ABQ 000 011 101 111

Combine gates.  Gates can be combined.  The output of one gate can become the input of another.  Try to determine the logic table for this circuit.

Construct the logic table for these circuits.

What happens when you add a NOT to an AND gate? ABQ 001 011 101 110 ABQ 000 010 100 111  Not

Can you make a NOT gate from an NAND? AQ 01 10

Can you make an AND gate from an NAND? ABQ 000 010 100 111

Can you make an OR gate from an NAND? ABQ 000 011 101 111

All computations can be done with an NAND gate.  The most sophisticated computer can be made with NAND gates from Radio Shack.  Need millions of them.

How do we use gates to add two binary numbers?  Binary numbers are either 1 or 0, either on or off.  Have two outputs.  Need a gate to produce each output. 0011 + 0+ 1+ 0+ 1 0001 10 ABQCO 0000 0110 1010 1101

How do I add larger numbers?  You can write any number in binary.

Add binary numbers.  Adding larger number just adds more columns.

Gang circuits in sequence.  To construct a full adder circuit, we'll need to gang these simple circuits together.  The “carry” value will become an input for the subsequent  Now have three inputs and two outputs.  Since we'll have both an input carry and an output carry, we'll designate them as C IN and C OUT.

Different Version of Full Adder.  Now we can add two binary bits together  accounting for a possible carry from the next lower order of magnitude.  sending a carry to the next higher order of magnitude. One-bit Full Adder with Carry-In and Carry-Out CIABQCO 00000 00110 01010 01101 10010 10101 11001 11111

Add multiple bits.  A full adder must be allocated for each bit to be added simultaneously.  To add two 4-bit numbers to produce a 4-bit sum (with a possible carry), you would need four full adders with carry lines cascaded.  By extension, two binary numbers of any size may be added in this manner.  http://www.play-hookey.com/digital/adder.html

Gates from electrical relays.  NOT gate  AND gate  OR gate

Test your knowledge on Boolean logic.  http://educ.queensu.ca/~compsci/resources/BoolLogic /assocdistrib.html

Here is a great resource for digital electronics.  http://hyperphysics.phy- astr.gsu.edu/Hbase/electronic/digcktcon.html#c1

Following are extra slides which may not be used.

Binary numbers.

“Exclusive” gates  Exclusively OR gate are true if either input is true but not both. ABQ 000 011 101 110 ABQ 001 010 100 111

Write out the logic tables for these two circuits.

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