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Math 4389compact summary of an undergra
Abstract Algebra
These notes are intended to be ad recall:rmaesugmbaerstsaom number that’s a multiple of 4.
Divisibiltthe:ehstsdmgiaital,oidt,is aormutiple of 3.mnu1isaiviiliel,b ful:
.
f and the TBt Ε
, then of
d n uade
SA  and for some
of
and
Sy Sx 1T
f image inverse image if is odd
1 if is even
fA B xx f f
y b n e v i g f fx Determine 0,1,5,9 and
For yDour:referis called thecalled theinitions you may find help
1 1 1
f f 2 f f
: given by : given by
Ex: f fx x Determine 7, 1,0,1,2 Ex: f fx 7,D1,r,1,ne 2, 1,0,1,2 and one
as inverse
, if and only if
, if and only if
injective
surjective
, or
onto
onetoone
example, every singleton set h
fs called
et or a set of size 2? fis called
.
. Then
:
fA B
. The function
fA
Is ittutisteit,ir theeprevtoDef: Let x A fucorrespondence, or bijective.onetoone is called a one to fonto? fonto?
fonto? fonto?
fonetoone? I fsnetoone? Is
fonetoone? I fsnetoone? Is
. Is . Is 2. Is 3 . Is
3 2
xx xx
x x f f
f f
be given by be given by
be given by be given by
f f f f
Ex: Let Ex: Let Ex: Let Ex: Let fonto? fnto?
fonetoone? Is fnetoone? Is
. Is
. Is
xx xxx
f f
be given by be given by
: :
f f
Ex: Let Ex: Let .
x x
if is evens odd
3
x 2 2
fx
onto?
f
be given by
f
onetoone? Is
f
Ex: Let Is nve sifi xx
xx 2 if is odd
be given by
gx
: . Problems? Why?
g
and
and
x and
if is even
: fg gf
f ddoxs 2 fi
x
fx
Ex: Let Compute ,
a and
of 
ca
with
multiple c
if bhere is an inteber denoted by
b
of and
is an integer
br that b ab otherwise
divisor
and 
is a a (gcd) of
a aivides
and
0 f 0 i 
. and is defined by
 of inteers cb
ab c
,
y max :
bare integers, then
. We also say that
and greatest coabon divisor
a b = ad common ivisor
ab ab
. Th
Number T DheforIfReview:enDef: Ais by , or simpl p
1and the only divisors of
m not 0. Then there exists a
p
if
dcan be wridis the smallest positive integer that can
, and
. Also, n
b prime integer
and is a
aand m p
p
be integers, at least one of the
b b,
and
aand and
a
TheoLemunique gcd Common Divisor):einfire.er .

pb
or
pa
, then either

pa
is prime and
p
Theo Ifem (Euclid’s Lemma) l to the
of two integers is always equa
GCD Fan:rduMprdfamtplGrLhMmndfves0 and 3234? by writing
.
ab
:
m
divides
m
if
od
ab m
Congruence Modulofine an equivalence relation on the set ,
.
d o m
abm
relation (reflexive, symmetric
is the set
Nottrahsatthi).is an equivaTenecequivalence class q
.
.
and if and only if they have the same
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