# If S has the same number of elements as T, then T has the same number as S...

• Jul 21st 2012, 09:59 AM
VonNemo19
If S has the same number of elements as T, then T has the same number as S...
Hi. The proof that I'm working is in the title. The Defition of "same number as" is given as follows: $(1)$Two sets $S$ and $T$ are said to have the same nuber of elements if each element of $S$ can be paired with precisely one element of $T$ in such a way that every element of $T$ is paired with precisely one element of $S$.

Notation: If $\pi$ is a pairing of the elements of $S$with with those of Tand the element sof S is paired in $\pi$ to the element $t$of T $,$ we shall write $s\overbrace{\leftrightarrow}^{\pi}{t}$ (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 $S$ has the same number of elements as $T$, we can select a pairing between the elements of $S$and $T$ in accordance with $(1).$We define a pairing as follows: If $s$ is paired with $t$ by the selected pairing $\pi$, then pair $t$ with $s$. That is if $s\overbrace{\leftrightarrow}^{\pi}{t}$, then $t$ is paired with $s$ to form the pairing of the elements $T$ with those of $S$. If the original pairing satisfied $(1),$then so will the new pairing. Specifically, since $\pi$had each element $T$ paired with a unique element of $S$, then the second pairing also has this property. Therefore, $T$ has the same number of elements as $S$.

Please help me to understand why this proposition is not trivial, and also the procedure of the proof.
• Jul 21st 2012, 10:34 AM
Plato
Re: If S has the same number of elements as T, then T has the same number as S...
Quote:

Originally Posted by VonNemo19
Hi. The proof that I'm working is in the title. The Defition of "same number as" is given as follows: $(1)$Two sets $S$ and $T$ are said to have the same number of elements if each element of $S$ can be paired with precisely one element of $T$ in such a way that every element of $T$ is paired with precisely one element of $S$.
Notation: If $\pi$ is a pairing of the elements of $S$with with those of Tand the element sof S is paired in $\pi$ to the element $t$of T $,$ we shall write $s\overbrace{\leftrightarrow}^{\pi}{t}$ (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 $S$ has the same number of elements as $T$, we can select a pairing between the elements of $S$and $T$ in accordance with $(1).$We define a pairing as follows: If $s$ is paired with $t$ by the selected pairing $\pi$, then pair $t$ with $s$. That is if $s\overbrace{\leftrightarrow}^{\pi}{t}$, then $t$ is paired with $s$ to form the pairing of the elements $T$ with those of $S$. If the original pairing satisfied $(1),$then so will the new pairing. Specifically, since $\pi$had each element $T$ paired with a unique element of $S$, then the second pairing also has this property. Therefore, $T$ has the same number of elements as $S$. Please help me to understand why this proposition is not trivial, and also the procedure of the proof.

I agree it is trivial from my point of view. But without seeing your notes, I cannot sure of that.
If $S$ has the same same number of elements as $T$ there is a bijection $\pi:S\to T.$
But $\pi^{-1}:T\to S$ is also a bijection.
• Jul 21st 2012, 01:10 PM
HallsofIvy
Re: If S has the same number of elements as T, then T has the same number as S...
The crucial part that a bijection is invertible. If f is a bijection from A to B, then its inverse is a bijection from B to A.