Is there a common notation for enumerating a set? For example if $A=\{2,4,6,\ldots,n \}$ is the set of even numbers, I would like to know the notation that enumerates ordered pairs $(e,i) \in \operatorname{enumerate}(A)$, where $e$ is the $i$:th element of $A$. So the sequence would be $(2,1),(4,2),\ldots,(n,m)$.
Would $(e,i) \in \operatorname{enumerate}(A)$ be ambiguous or is there a more common way of describing this kind of enumeration?
Best Answer
Normally, you would just write $A = \{a_i : i \in \omega\}$ or $A = \{a_i : i \leq n\}$ where $a_i = 2i$.
Of course a sequence such as $(a_n)$ is defined to be a function; the sequence of pairs you have described is the graph of the function $a$. We could write $a(i)$ instead of $a_i$. And, in the usual set theory way, we can identify $a$ with its graph, so that $a$ is a set of ordered pairs. But it is more common to write $a_i = 2i$ than it is to write $(i,2i) \in a$. Particularly outside of set theory, it is more common to treat functions as a kind of first-class object than to treat them as sets of ordered pairs.