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20. NOVEMBER 15TH: CARDINALITY II
47
20. November 15th: Cardinality II
20.1
.
Give an alternative proof of Lemma 20.1 by proving directly that the func
tion
f
:
A
→
B
deFned in the proof given in
§
20.1 is both injective and surjective
(without using the inverse functions
g
S
).
Answer:
Recall that
f
is deFned by setting (for all
a
∈
A
)
f
(
a
)=
f
S
(
a
), where
S
is the unique element of the partition
P
A
containing
a
.
To prove that
f
is injective, assume that
f
(
a
)=
f
(
a
°
)fo
rtwoe
l
em
en
t
s
a
,
a
°
of
A
.Inp
a
r
t
i
cu
l
a
r
,
f
(
a
)and
f
(
a
°
)mustbe
longtooneandthesamee
lemento
fthe
partition
P
B
;ca
l
lth
i
se
lemen
t
T
.S
i
n
c
e
α
is a bijective map
P
A
→
P
B
,
T
is the
image of a unique element
S
of the partition
P
A
.Byth
eway
f
is deFned,
a
and
a
°
must both belong to
S
.S
in
c
e
f
S
is bijective, it is particular injective, and it follows
that
a
=
a
°
.Th
u
s
,w
eh
a
v
esh
ownth
a
t
f
(
a
)=
f
(
a
°
)imp
l
ies
a
=
a
°
,andth
isshows
that
f
is injective.
To prove that
f
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 Fall '11
 Aluffi

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