Hi. The proof that I'm working is in the title. The Defition of "same number as" is given as follows:
Two sets
and
are said to have the same number of elements if each element of
can be paired with precisely one element of
in such a way that every element of
is paired with precisely one element of
.
Notation: If
is a pairing of the elements of
with with those of Tand the element sof S is paired in
to the element
of T
we shall write
(I don't know how to put the \pi above the \leftrightarrow without the \overbrace).
I don't really understand the book's proof: Since
has the same number of elements as
, we can select a pairing between the elements of
and
in accordance with
We define a pairing as follows: If
is paired with
by the selected pairing
, then pair
with
. That is if
, then
is paired with
to form the pairing of the elements
with those of
. If the original pairing satisfied
then so will the new pairing. Specifically, since
had each element
paired with a unique element of
, then the second pairing also has this property. Therefore,
has the same number of elements as
. Please help me to understand why this proposition is not trivial, and also the procedure of the proof.