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.
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.
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. 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.
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 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.
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
“Exclusive” gates • Exclusively OR gate are true if either input is true but not both.