# 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: $\displaystyle (1)$Two sets $\displaystyle S$ and $\displaystyle T$ are said to have the same nuber of elements if each element of $\displaystyle S$ can be paired with precisely one element of$\displaystyle T$ in such a way that every element of$\displaystyle T$ is paired with precisely one element of $\displaystyle S$.

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

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