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.