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 nuber 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.