Download - Last that if S an
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9 3 21 ColleetHW
Last time showed that if S has an identity element it'sunique
the let S l be a bin structure w identity elf e let s l be
another binarystructure Assume thereis a ftu d s s satisfying
thehomem property and assume d is surjective onto Then Ole is an
identity element for shence the identityelement for s
if ur x e s Wis del x x te del x
Assume x 0 x where x is some il't of S ok since 0 is out
Ole x del out Ole x 01 1 x
The other is thesame11
See examples at end of 53
4 GroupFormalize some propertiesof R or I or Q or I that we
take for granted
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Ex Solve for x x 3 8
x 3 G 8 1 3 add 3 to bothsides
7 31 63 5 computation
7 3 61 5 assoc3 is theadditiveinverseof 3
y 5 property of 0
thisworksble Rt is agaspmi partie
G'G G
Def Aged G l is a binary structure satisfy
Cal f a b c e G Ca b c a btc assoc of
b J ee G st axe e a a fate identity elf
c foreach a EG 7 a e G st aka a a pinverses
I will bechangingthisnotation shortly both
and C
Note that even commutativityisn't assumed
Def A group is abelian if is commutative
Ey U all IR t and Z t abel
Un abel Rt all
abel Maciel t all
t abd Ghar not all
where Gh R is
29
E frombookI
let G at
Define a b by a tb of
Clan Cat is a group
a check assoc catb c e at cl last 94 a I
a b ax b c
h identify elf is 2
at 2 4 a 2 a
c Given a a Qt check a I theinverseelement
ax I I n tReward Don't gettoo dependent
on whatyouknow for familiar objects e.g
IR Q E We have a set a binaryoperation and a coupleofaxioms and
we want to know what we can say about it
Propertifgroups
the identity elt is unique just saw
the Clift rightcancellation
Gt G l be a group let a b c e G
If a b at c ther b c
n If b a c a then b c
30
pHlet a kegu
inverseof a
a b a c
a atb a ate
a a b Ca al c
ex b e c
b c
Same for ight cancellation X
TI let G l he a group let a be G Then
la the beer egg at x b has a soln namely x a b ohne a
is the inn of ab thesolution is uniquec Simbly y a b has a unique solufor y
pfal x a b is g solution plugin
b Uniqueness Say x Xz are solusThen a x b a z Leftcancellation
o samefor y a b k
thy for each at G its inverse a is unique