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

Axioms

1. x + 0 = x

2. x + 1 = 1

3. x + x = x

4. x + x’ = 1

5. x + y = y + x

6. x + (y + z) = (x + y) + z

7. x(y + z) = xy + xz

8. (x + y)’ = x’y’

9. (x’)’ = x

10. x · 1 = x

11. x · 0 = 0

12. x · x = x

13. x · x’ = 0

14. xy = yx

15. x(yz) = (xy)z

16. x + yz = (x + y)(x + z)

17. (xy)’ = x’ + y’

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)

Logic Circuits (Gates)

• 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

AND

OR

NOT

Schematic Symbols

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

AND

OR

BOR

A

CA

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!

A

CB

AND

OR

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

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?

Axioms/Gates

• Based on the axioms, the AND and OR gates may be created with more two inputs– Which axioms allow for this?

More Gates

• NAND– Shortened form of “not and”– Truth table?– Symbol

NAND

More Gates

• NOR– Shortened form of “not or”– Truth table?– Symbol

NOR

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)

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

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…

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