The CantorBernsteinSchr¨
oder Theorem
The following theorem is a precise formulation of the (obvioussounding) fact that if “
A
has
fewer elements than
B
” and “
B
has fewer elements than
A
”, then
A
and
B
are equinumerous.
Theorem (Bernstein 1897).
Let
A
and
B
be two nonempty sets. Assume that there exist
two onetoone maps
f
:
A
→
B
and
g
:
B
→
A
. Then, there exists a bijection
h
:
A
→
B
.
In fact, there are partitions
A
=
A
1
]
A
2
and
B
=
B
1
]
B
2
such that
f
restricted to
A
1
is a
bijection from
A
1
to
B
1
, and
g
restricted to
B
2
is a bijection from
B
2
to
A
2
.
Hence the function
h
(
x
) =
f
(
x
)
if
x
∈
A
1
g

1
(
x
)
if
x
∈
A
2
is a bijection from
A
to
B
.
http://en.wikipedia.org/wiki/CantorBernsteinSchroeder
theorem
http://en.wikipedia.org/wiki/Cardinal
number
http://www.math.rutgers.edu/ useminar/bernstein.pdf
Here’s a surprising application: Bernstein’s theorem implies that we can dissect a square
into two disjoint parts
S
1
and
S
2
and a circular disk into
C
1
and
C
2
in such a way that
S
1
is
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 Spring '08
 ctw
 Math, Finite set

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