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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. - PowerPoint PPT Presentation

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Page 1: BOOlean Logic

BOOlean Logic

The basis for digital electronics.

Page 2: BOOlean Logic

What digital electronics do you use?

Computer CD & DVD players IPod Cell phone HDTV Digital cameras

Page 3: BOOlean Logic

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

Page 4: BOOlean Logic

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.

Page 5: BOOlean Logic

Have you used Boolean logic?

What information have you searched for on the Web? Do a Google Search for car radios.

Page 6: BOOlean Logic

The AND operator (both, all)

rivers AND salinity

dairy products AND export AND Europe

Page 7: BOOlean Logic

The OR operator (either, any)

fruit OR vegetables

fruit OR vegetables OR cereal

from CSIRO Australia

Page 8: BOOlean Logic

The NOT operator

fruit NOT apples

Look at the Boolean Machine http://kathyschrock.net/rbs3k/boolean/

Page 9: BOOlean Logic

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.

Page 10: BOOlean Logic

What would a library search give for “stress” and “anxiety”?

Shaded areas indicate retrieval

Search Statement Types of records retrievedNumber of records retrieved

            

    

stress OR anxietyDocuments that discuss

either stress or anxiety1127

            

    

stress AND anxietyDocuments that discuss

both stress and anxiety51

            

    

stress NOT anxietyDocuments that discuss

stress but not anxiety687

Page 11: BOOlean Logic

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.

Page 12: BOOlean Logic

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.

Page 13: BOOlean Logic

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.

Page 14: BOOlean Logic

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

Page 15: BOOlean Logic

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.

A Q

0 1

1 0

From How Stuff Works

Page 16: BOOlean Logic

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).

A B Q

0 0 0

0 1 0

1 0 0

1 1 1

Page 17: BOOlean Logic

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

A B Q

0 0 0

0 1 1

1 0 1

1 1 1

Page 18: BOOlean Logic

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.

Page 19: BOOlean Logic

Construct the logic table for these circuits.

Page 20: BOOlean Logic

What happens when you add a NOT to an AND gate?

A B Q

0 0 1

0 1 1

1 0 1

1 1 0

A B Q

0 0 0

0 1 0

1 0 0

1 1 1

Not

Page 21: BOOlean Logic

Can you make a NOT gate from an NAND?

A Q

0 1

1 0

Page 22: BOOlean Logic

Can you make an AND gate from an NAND?

A B Q

0 0 0

0 1 0

1 0 0

1 1 1

Page 23: BOOlean Logic

Can you make an OR gate from an NAND?

A B Q

0 0 0

0 1 1

1 0 1

1 1 1

Page 24: BOOlean Logic

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.

Page 25: BOOlean Logic

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.

0 0 1 1

+ 0 + 1 + 0 + 1

00 01 01 10

A B Q CO

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

Page 26: BOOlean Logic

How do I add larger numbers?

You can write any number in binary.

Page 27: BOOlean Logic

Add binary numbers.

Adding larger number just adds more columns.

Page 28: BOOlean Logic

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 CIN and COUT.

Page 29: BOOlean Logic

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

CI A B Q CO

0 0 0 0 0

0 0 1 1 0

0 1 0 1 0

0 1 1 0 1

1 0 0 1 0

1 0 1 0 1

1 1 0 0 1

1 1 1 1 1

Page 30: BOOlean Logic

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

Page 31: BOOlean Logic

Gates from electrical relays.

NOT gate

AND gate

OR gate

Page 32: BOOlean Logic

Test your knowledge on Boolean logic.

http://educ.queensu.ca/~compsci/resources/BoolLogic/assocdistrib.html

Page 33: BOOlean Logic

Here is a great resource for digital electronics.

http://hyperphysics.phy-astr.gsu.edu/Hbase/electronic/digcktcon.html#c1

Page 34: BOOlean Logic

Following are extra slides which may not be used.

Page 35: BOOlean Logic

Binary numbers.

Page 36: BOOlean Logic

“Exclusive” gates

Exclusively OR gate are true if either input is true but not both.

A B Q

0 0 0

0 1 1

1 0 1

1 1 0

A B Q

0 0 1

0 1 0

1 0 0

1 1 1

Page 37: BOOlean Logic

Write out the logic tables for these two circuits.