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Why?

Why?. What’s Boolean algebra used for? “The purpose of Boolean algebra is to facilitate the analysis and design of digital circuits.” Express a truth table relationship in algebraic (symbolic) notation Express a logic diagram’s input/output relationships in algebraic notation

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  1. Why? • What’s Boolean algebra used for? • “The purpose of Boolean algebra is to facilitate the analysis and design of digital circuits.” • Express a truth table relationship in algebraic (symbolic) notation • Express a logic diagram’s input/output relationships in algebraic notation • Provides a means for finding simpler circuits to implement a given function

  2. x + 0 = x x + 1 = 1 x + x = x x + x’ = 1 x + y = y + x x + (y + z) = (x + y) + z x(y + z) = xy + xz (x + y)’ = x’y’ (x’)’ = x x · 1 = x x · 0 = 0 x · x = x x · x’ = 0 xy = yx x(yz) = (xy)z x + yz = (x + y)(x + z) (xy)’ = x’ + y’ Axioms

  3. Axioms • 1, 10 – identity laws • 2, 11 – one, zero laws • 3, 12 – idempotence laws • 4, 13 – inverse laws • 5, 14 – commutative laws • 6, 15 – associative laws • 7, 16 – distributive laws • 8, 17 – De’ Mogan’s laws • 9 – double negation law (?? – I made that name up)

  4. AND OR NOT Logic Circuits (Gates) Schematic Symbols • These are the things computers (and other digital devices) are made of • Circuit designers use Boolean algebra to design circuits drawn on schematic drawings • Fabrication facilities use schematic drawings to produce silicon chips

  5. AND OR OR A B A C Axioms • Axioms are one way to simplify a Boolean expression • Consider the expression F(A,B,C) = (A+B)(A+C) • Requires 3 logic gates

  6. AND OR A B C Axioms • Using the axioms one can “prove” that this expression simplifies F(A,B,C) = (A+B)(A+C) = A + BC (this is axiom 14 but it can be proved from the others) • Resulting in a two gate solution • We shaved 33% off of our design!

  7. Simplification • Why bother? • A simplified expression means a smaller circuit • A smaller circuit is easier to build • A smaller circuit is easier to debug • A smaller circuit means more money in your pocket

  8. Another Example • F(A,B,C) = ABC + ABC’ + A’C • How many gates are required as specified? • What does the circuit look like? • Can the expression be simplified based on the axioms? If so, how?

  9. Axioms/Gates • Based on the axioms, the AND and OR gates may be created with more two inputs • Which axioms allow for this?

  10. NAND More Gates • NAND • Shortened form of “not and” • Truth table? • Symbol

  11. NOR More Gates • NOR • Shortened form of “not or” • Truth table? • Symbol

  12. NAND/NOR • So, what’s so special about NAND and NOR? • NAND and NOR are considered “universal gates” • That is, anything that can be done with AND/OR/NOT can be done with only NAND or NOR gates (one or the other, not both)

  13. NAND/NOR • The universality of NAND/NOR is important because it means you can make many copies of a single gate type on a single piece of silicon and then use it to create complex circuits on a single chip • Programmable (FPGA) devices use this technology

  14. Simplification • Using the axioms to “prove” that a simplified version of a circuit is equivalent to the complex version takes a special kind of person… • …of which I’m not one • Fortunately, there’s another way…

  15. Homework • Chapter 1 • 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 1-7 • Create an AND gate using only NAND gates • Create an AND gate using only NOR gates • Create an OR gate using only NAND gates • Create an OR gate using only NOR gates • Create a NOT gate using only NAND gates • Create a NOT gate using only NOR gates • Due next lecture

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