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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 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.

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