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Back to 2.3 : functions.
Roell thot f: ASB is a function if f assigns
flatE B to each ae A. Ais domain , Bis codomain.
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Two very important properties a fanetai they
( t ) f is injective iif intuitively, it separates elements of Ain B.
Precisely ,
injective ,
O:-# puppet injective.
Linear like flick are functions 2×+1
To prove this , fix arbitrary a. , as C- IR .
flat Eflazl
have a amc- A’ (a.Far → flattflazl
-a.far→ flattflail Assuminganon- equality prove .
is very unnatural , so we almost always proceed by
contrapositive , so assume
1 29=29 so
La 1-I -‘ so , ,
Polynomial
functions that aren’t
a.Faz a flat flan .
← f’s graph dog horizontal the
Ha, , az Fa, .az
injective,
T (a.Far→ flatEffort)
glock 3×4- x
is not (9. Far
w e must flail Ffkz) )
) flan prove ,=az
fla – ) $ a
gift ) – 2
gut – and g. 111=4 > 2 , and g is
text – x42
– 3) i. hall
(2) Surjective
thatif f:A→B,
therangeoffis
He set of all robes fan
NT ( Wka a e
is not ‘ve
inject ‘re since
constructive
Hat-2 ÷ i ,
continues , suehtletglal-2.at 0 , so
‘ = 0 if x= ‘T
In general, flat FB .
If flat-43 , Hen ve surjective
surjective
Leis ? f-(A1 EB
BE f-CAI , f-flat for sore
if AB That
theB FaeA ( A
surjective
flat -_ definition
prove surjective find an a
arbitrary 6 ,
let’s show flat —
2x-it is surjective f: Rsk ,
at IR with flat This a-
Hard to do for border functions!
glx) = x’txt’d
– the flat-
, through every vote .
continuous.
surjective ! ( Non –
constructive ! )
range FLA#B, so
w e know tin
XP . Thus g
surjective .
surjective ? for
instance ,
7-beBHaeA flat#b
but not dykes’ve,
there functions vice versa ?
IR l l’t Kees→
tan : surjective but not
h64- I’x hIR-{03→IRisinject.’re
surjective f
surjective,
Similarly ,
I¥÷÷:÷t¥÷÷::÷÷:÷: : .
to-I× V K :÷:÷¥÷.
lo – l LO – ) ✓ ‘:b if 670
, V X effing -r
so squaring is surjedbeforcodonein [0,x).
different versions
f. G e t.fi/–fzlxkfdxkx2
Last time , Reali tht
surjective
f injective
live, due end of
Homework 4
EOD tomorrow
Four m o re to finish
after this Chester
introduced
surjective
: flex day .
fled flu X)
HyeY F* X (flocky).
Examples often Ask lest tim, as with most functions from calculus .
functions :
μ. is a iinkforan
The Star Loch is
L j : R S I the greatest integer leg Hon
max ({me Elmzx}
,- is the set of
L 2.781=2, L- it ) = –
that Lil is surjective : for any ne E n – Ln)
injective :
of Max , if
117.51–1171=17
egad to x .
all a finite
( surjective, not injective)
all assets of Rohit
then m a x
tersest element
not ihjeethe.
eberts of A
. here a largest element {meIluz#3, but not g B or
finite sets, 10,1) ),
A’s N 01004=5
to itsleft
Compositions
DEA . Genedly,
Hot case we
he’ll peeing consider fog when D=A . can draw a nice diagram :
composition
fandoms fig
fog late ffgbdl.
all functions
composed I olalde
, f: ASB $ g.CSD
holgof) echog)of
g : Bsc, @g) of Cal
Ismene:*:c:*:.mn: :*: are associative so is composition :
Composition
associative
holgofkal ‘
if 8 A Srp
B → PII We must prove
h (go hfg (flat)
i.e . f. ASB,
h. (gof)lat – =
hog (flat)
(hog) of La) is desired. DE
W e t d a t
We’d like goff
Het is to °
as O is to t end 1 is to x
oomnoteuily ?
such = Afp.
g :B- C I Bbc
for my f: A→ B, Thisimplies:BSo
a do – BSBbylight- bforall*e-B
43 : lista
This is called
composition: if
the idalfg function , douse it is the identity f: A- B, then four = f= 4gof
a exists we call ,g
:A→B If such A
This means f-(gli) ) – I,gCb)
‘ fight =D
He cstketeai of ad sets ad old farting with the composition portion, is called ,
↳Wet about incenses ? xttxko , x. ‘he:L Howabout fog=3got-‘ la?
inane function
and . “andlax
function, b e t ,
e x i s t s a ‘lb) ”
f- f-‘ ‘ loll is the only possible solution.
namely f-‘
f-( F’1611=6
definition .
f-‘ (flail f”(61
proof !) fitful
And (uniqueness
– a,bydef -.
fat- b Kade. –
f is injective
solution to –
surjective :
if flat flash
f-‘ heal) f ‘Ifad )
So, a function with an inverse must be injeefie § surjective ! (Bijective)
Conversely, if f is bijective , then it les an house . Bijective a s invertible !
i n injective n o surjecfie
{1,23> {42,33
{ 112,333 91,23
Missed? I .2
!! 3 • nowhere else
to godo This is baack
191,4331 > I22
generalizes : and
injections
sarjeefiog
A S B i f ASB, Hon
I A K I B I
Hl > 1131 ”
In particle, if
there is a
IAHBligre.g.pe,
IBI . Two sets can only fare
a bijection
between ten
cardinality .
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