Schroeder-Bernstein Theorem
The Schroeder-Bernstein Theorem (sometimes called the Cantor-Schroeder-Bernstein Theorem) is a result from set theory, named for Ernst Schröder and Felix Bernstein. Informally, it implies that if two cardinalities are both less than or equal to each other, then they are equal.
More specifically, the theorem states that if and
are sets, and there are injections
and
, then there is a bijection
.
The proof of the theorem does not depend on the axiom of choice, but only on the classical Zermelo-Fraenkel axioms.
Proof
We call an element of
lonely if there is no element
such that
. We say that an element
of
is a descendent
of an element
of
if there is a natural number
(possibly zero)
such that
.
We define the function as follows:
Note that
cannot be lonely itself. If
is the descendent of a lonely point, then
for some
; since
is injective, the element
is well defined. Thus our function
is well defined.
We claim that it is a bijection from
to
.
We first prove that is surjective. Indeed, if
is the
descendent of a lonely point, then
; and if
is not
the descendent of a lonely point, then
is not lonely, so there
is some
such that
; by our definition, then,
. Thus
is surjective.
Next, we prove that is injective. We first note that for any
, the point
is a descendent of a lonely point if and only
if
is a descendent of a lonely point. Now suppose that we have
two elements
such that
. We consider
two cases.
If is the descendent of a lonely point, then so is
.
Then
. Since
is a
well defined function, it follows that
.
On the other hand, if is not a descendent of a lonely point,
then neither is
. It follows that
.
Since
is injective,
.
Thus is injective. Since
is surjective and injective, it is
bijective, as desired.