1. Computer Systems 1 Fundamentals 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
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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 ( )
“Against logic there is no armor like ignorance.”
Laurence J. Peter ( )
“A page of history is worth a pound of logic.”
Oliver Wendell Holmes Jr. ( )
“Logic is like the sword--those who appeal to it shall perish by it.”
Samuel Butler ( )
“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 ( )
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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
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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
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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
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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
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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
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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
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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
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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
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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
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13. Simple Logic Circuits What’s the truth table? E.g.-
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14. Simple Logic Circuits What’s the truth table? E.g.- OUTPUT= A+B + A•B
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15. NAND Gate NAND Two or more inputs
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
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16. NOR Gate NOR Two or more inputs
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 output becomes true
Boolean expression for NOR gate with 2 inputs (X NOR Y):
X+Y
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17. XOR Gate XOR Two or more inputs
Exclusive OR
Sometimes EOR
Two or more inputs
Inputs must be different to produce a true output
Both inputs being true or false result in a false output
Boolean expression for XOR gate with 2 inputs (X XOR Y):
X•Y + X•Y
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18. XNOR Gate XNOR Two or more inputs
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 false result in a true output
Boolean expression for XNOR gate with 2 inputs (X XNOR Y):
X•Y + X•Y
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19. Other Logic Circuits Simplify last circuit using NOR gate Becomes:
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20. Other Logic Circuits E.g.- OUTPUT= X•Y + (Y•Z + Y•Z) X Y Z
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21. Other Logic Circuits X Y Z
E.g.-
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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 ( )