How to verify if an element is inside an ordered pair?

Question

The notation to verify if an element belongs to a set is $e \in E$. But which notation should I use to verify if an element is part of an ordered pair? Is $a \in (a,b)$ valid (e.g., $1 \in (1,2)$)?

Thanks!

Answer 1

As I stated in my comment above, I am not aware of any accepted standard notation for saying a particular element appears in an ordered pair.

That said, suppose we take the view that order $n$-tuples are "really" maps from $\mathbb{Z}_{>0} = \{1,2,3,\dots\}$ into our set $X$. Or more generally from an ordinal into $X$. (This sort of thing is done in field such as Topology where one runs into infinite Cartesian products).

From this viewpoint: $(a,b)=f$ where $f:\{1,2\} \to X$ and $f(1)=a$, $f(2)=b$. If we take this view $a, b \in \mathrm{Range}(f)$. So we could write $1 \in \mathrm{Range}(1,2)$.

However, try doing that without explaining yourself and you'll get some odd looks. :)