Computer Systems 1 Fundamentals of Computing

1. Computer Systems 1Fundamentals of Computing Computer Logic

2. CS1: Week 18 • What’s Logic? • Truth Tables • Simple Logic Gates • Simple Logic Circuits • Other Logic Gates • Other Logic Circuits Computer Systems 1 (2004-2005)

3. What’s Logic? • “The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.” • Albert Einstein (1879 - 1955) • “Against logic there is no armor like ignorance.” • Laurence J. Peter (1919 - 1988) • “A page of history is worth a pound of logic.” • Oliver Wendell Holmes Jr. (1841 - 1935) • “Logic is like the sword--those who appeal to it shall perish by it.” • Samuel Butler (1835 - 1902) • “Somebody who thinks logically is a nice contrast to the real world.” • The Law of Thumb • “Insanity is often the logic of an accurate mind overtaxed.” • Oliver Wendell Holmes (1809 - 1894) Computer Systems 1 (2004-2005)

4. It’s Logical, Captain... • Logic is concerned with conditions • Conditions are either achieved or not achieved • There are two states used within logic: • True • False • This could also be thought on as: • ON and OFF • 1 and 0 • YES and NO Computer Systems 1 (2004-2005)

5. It’s Logical, Captain... • Logic is used in a variety of situations, most importantly: • Real life situations • Inside the computer • To perform logical and arithmetic functions • Real life logical situations: • There’s a buzzer in your car that sounds when the headlights are on and the door is open • The fire alarm installed in your home will go off if it senses heat or smoke • If I’m not tired then I will go to the pub tonight Computer Systems 1 (2004-2005)

6. Logic Gates • Each logic gate has it’s own features: • Symbol • Boolean Algebraic Expression • Truth Table • Gates can be used to build logic circuits • Nothing to do with Bill GATES • Even more thankfully, nothing to do with Gareth GATES • Each gate has a number of inputs and outputs • Usually multiple inputs and ONE output • Circuits can be created to perform situation testing and produce and output Computer Systems 1 (2004-2005)

7. Truth Tables • Truth tables are used to represent the functionality of a logic gate or circuit • Truth tables are constructed by analysing all possible combinations of values that can be sent to a logic gate or circuit • All possible outputs are then calculated • Truth tables allow us to show the functionality of a logic gate or circuit • We can also derive expressions and simplify complex circuits by analysing the truth tables • More on that next week Computer Systems 1 (2004-2005)

8. Boolean Algebra • Named after George Boole • English Mathematician • Provides a method to express functions and transforms using logical variables • Commonly letters of the alphabet • A, B, D, X, Y, Z, etc. • E.g.- X + Y = Z • Logic gates and circuits work on the principles of Boolean logic • In the computer TRUE or FALSE is represented by a high or low voltage • 1 or 0 Computer Systems 1 (2004-2005)

9. AND Gate • Two or more inputs • All inputs must be true to produce a true output • E.g.- A AND B must be true • All other combinations between inputs result in a false output • Boolean expression for AND gate with 2 inputs (X AND Y): • X•Y Computer Systems 1 (2004-2005)

10. OR Gate • Two or more inputs • At least one input must be true to produce a true output • E.g.- A OR B must be true • Both inputs being true result in a true output • Boolean expression for OR gate with 2 inputs (X OR Y): • X+Y Computer Systems 1 (2004-2005)

11. NOT Gate • Usually only one input • The value of the input is inverted • TRUE becomes FALSE • FALSE becomes TRUE • E.g.- A is NOT true/false • Boolean expression for OR gate with 1 input(X): • X • Sometimes ~X Computer Systems 1 (2004-2005)

12. Simple Logic Circuits • Circuits comprise of one or more logic gates • Gates are joined together • Usually the process flow moves left to right • Truth tables can be constructed for circuits • Helped by deriving Boolean algebra expressions for gates and the output of the circuit • Circuits are used to construct useful logical processes • The computer (CPU) is a complex logic circuit Computer Systems 1 (2004-2005)

13. Simple Logic Circuits • E.g.- What’s the truth table? Computer Systems 1 (2004-2005)

14. Simple Logic Circuits • E.g.- • OUTPUT= A+B + A•B What’s the truth table? Computer Systems 1 (2004-2005)

15. NAND Gate • NAND • NOT AND • Two or more inputs • If all inputs are true then output is false (0) • All other combinations between inputs result in a true output • Boolean expression for NAND gate with 2 inputs (X NAND Y): • X•Y Computer Systems 1 (2004-2005)

16. NOR Gate • NOR • NOT OR • Two or more inputs • At least one input must be true to produce a false (0) output • If both inputs are false then outputbecomes true • Boolean expression for NOR gate with 2 inputs (X NOR Y): • X+Y Computer Systems 1 (2004-2005)

17. XOR Gate • XOR • Exclusive OR • Sometimes EOR • Two or more inputs • Inputs must be different to produce a true output • Both inputs being true or falseresult in a false output • Boolean expression for XOR gate with 2 inputs (X XOR Y): • X•Y + X•Y Computer Systems 1 (2004-2005)

18. XNOR Gate • XNOR • Exclusive NOR • Two or more inputs • One input must be true to produce a false output and both inputs must be different • Both inputs being true or falseresult in a true output • Boolean expression for XNOR gate with 2 inputs (X XNOR Y): • X•Y + X•Y Computer Systems 1 (2004-2005)

19. Other Logic Circuits • Simplify last circuit using NOR gate • Becomes: Computer Systems 1 (2004-2005)

20. X Y Z Other Logic Circuits • E.g.- • OUTPUT= X•Y + (Y•Z + Y•Z) Computer Systems 1 (2004-2005)

21. X Y Z Other Logic Circuits • E.g.- Computer Systems 1 (2004-2005)

22. Do you know anything now? • What’s Logic? • Conditions • Truth Tables • Simple Logic Gates • AND • OR • NOT • Simple Logic Circuits • Other Logic Gates • NAND • NOR • XOR • XNOR • Other Logic Circuits Computer Systems 1 (2004-2005)