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Logic Gates

Logic Gates

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Logic Gates

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  1. Logic Gates Informatics INFO I101 February 3, 2003 John C. Paolillo, Instructor

  2. Items for Today • Last week • Digitization • Graphic, audio and video data sizes • This time • Digital logic, Boolean algebra, and circuits • Logic gates and truth tables

  3. Set Theory&Logic BooleanAlgebra ElectricCircuits Union Intersection Complement OR AND NOT Parallel Series Inversion Binary Arithmetic Addition Multiplication Twos Complement A Fundamental Unity

  4. + 0 1  0 1 0 00 01 0 0 0 1 01 10 1 0 1 Binary Arithmetic Means zero, carry 1(two distinct outputs)

  5. + 0 0 1 1  0 1 0 0 0 0 1 1 0 0 0 Exclusive OR 1 1 1 1 0 1 1 0 1 Boolean Algebra AND OR Binary addition:ExOR (for one’s place) plus AND (for carry)

  6. + off on  off on off off on off off off on on on on off on Electric Circuits Parallel Series

  7. A Fundamental Unity • Any logical or mathematical problem may be formulated in set theoretic terms • Boolean logic may be used to represent any set-theoretic operation • The fundamental elements of Boolean logic may be implemented in electric circuits • Therefore, any logical or mathematical problem may be implemented in electrical circuits!

  8. Logic Gates

  9. What are Logic Gates? • Logic gates are conventional arrangements of switches that are treated as units • They abstract away from the details of individual switches • They may be implemented in any number of ways • Only their function is important

  10. OR AND XOR Not Buff Symbols for Logic Gates

  11. A ¬A 0 1 1 0 xor nand or xnor nor and 0 0 0 0 0 0 1 1 1 1 1 1 A Buff 0 0 0 0 0 0 1 0 1 0 0 1 0 0 1 0 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 1 0 0 1 1 0 0 0 Functions of Logic Gates

  12. Boolean Logic Hints AND: if you know that one input is 0, it doesn’t matter what the other input is, the output will be 0 OR: if you know that one input is 1, it doesn’t matter what the other input is, the output will always be 1 XOR: if you know that the inputs are the same, then the output will always be 0

  13. Some Example Circuits

  14. A C E D A 0 1 0 1 B 0 0 1 1 C 0 1 1 1 D 1 1 1 0 E 0 1 1 0 B Synthesizing XOR

  15. A C E B D A 0 1 0 1 B 0 0 1 1 C 1 0 1 0 D 1 1 0 0 E 0 1 1 1 Synthesizing OR from NAND

  16. 01 11 r1 row 00 10 r2 c1 c2 black red col Encoding Position in a 2-by-2 Board

  17. col 01 11 r1 00 10 r2 c1 c2 row Decoding Position to a 2-by-2 Board

  18. d c C b A 0 1 0 1 B 0 0 1 1 C a b c d a A B 4-line multiplexer

  19. d c b A 0 1 0 1 a IN 0 0 0 b 0 IN 0 0 c 0 0 IN 0 d 0 0 0 IN B 0 0 1 1 a A B IN 4-line demultiplexer

  20. A E D C B A 0 1 0 1 B 0 0 1 1 C 1 0 1 0 D 1 1 0 0 E 0 0 0 1 Synthesizing AND from NOR

  21. Other Applications of Boolean Logic

  22. Graphic Paint/Copy Modes COPY OR XOR

  23. Database Search Queries • A keyword represents the set of documents containing it • Boolean AND represents the intersection of the sets corresponding to two (or more) search terms • Boolean OR represents the union of the sets corresponding to two (or more) search terms