BOOlean Logic

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# BOOlean Logic - PowerPoint PPT Presentation

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.

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## BOOlean Logic

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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,
• 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.
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 ORwhen 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.
• 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.

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).
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
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.
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.
How do I add larger numbers?
• You can write any number in binary.
• 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 CIN and COUT.
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.
• 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.