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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, • 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”?**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.