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