Logic Gates
description
Transcript of Logic Gates
Logic Gates
Informatics INFO I101
February 3, 2003
John C. Paolillo, Instructor
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
A Fundamental UnitySet Theory
&Logic
UnionIntersectionComplement
BooleanAlgebra
ORANDNOT
ElectricCircuits
ParallelSeries
Inversion
Binary Arithmetic
AdditionMultiplication
Twos Complement
Binary Arithmetic
+ 0 1
0 00 01
1 01 10
0 1
0 0 0
1 0 1
Means zero, carry 1(two distinct outputs)
Boolean Algebra
+ 0 1
0 0 1
1 1 1
0 1
0 0 0
1 0 1
OR AND
0 1
0 0 1
1 1 0
Exclusive ORBinary addition:ExOR (for one’s place)plus AND (for carry)
Electric Circuits
+ off on
off off on
on on on
off on
off off off
on off on
Parallel Series
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!
Logic Gates
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
Symbols for Logic Gates
AND OR
XOR Not
Buff
Functions of Logic Gates
and 0 1
0 0 0
1 0 1
or 0 1
0 0 1
1 1 1
xor 0 1
0 0 1
1 1 0
nand 0 1
0 1 1
1 1 0
nor 0 1
0 1 0
1 0 0
xnor 0 1
0 1 0
1 0 1
A ¬A
0 1
1 0
A Buff
1 1
0 0
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
Some Example Circuits
Synthesizing XOR
E
A0101
B0011
C0111
D1110
E0110
D
A C
B
Synthesizing OR from NAND
E
A0101
B0011
C1010
D1100
E0111
A C
BD
Encoding Position in a 2-by-2 Board
01 11
00 10
r1
r2
c1 c2
row
col
black
red
01 11
00 10
r1
r2
c1 c2
col
row
Decoding Position to a 2-by-2 Board
4-line multiplexer
d
c
b
a
C
A0101
B0011
CabcdA B
4-line demultiplexer
A B
A0101
B0011
aIN000
b0
IN00
c00
IN0
d000
IN
IN
d
c
b
a
Synthesizing AND from NOR
A0101
B0011
C1010
D1100
E0001
E
BD
A C
Other Applications of Boolean Logic
Graphic Paint/Copy Modes
COPY OR XOR
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