Notation for number of distinct elements in a set Let $L = \{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, ... ,a_{n}\}$ be a logtrace containing a finite set of antenna samples submitted within a time window.
What what would be a good way to express the number of distinct cellIDs $a^{cid}$ in $L$?
where $a_{t}^{cid}$ is the cellID of the current registered antenna at time $t$.
Assuming $A=\{1,4,3,5,4,3\}$, the result of the calculatin i'm searching for should be 4.
Btw: What would be $ \#(A \cap A)$ ? 4 or 6? (if the result is 4, i would still like to avoid this weird notation.)
Here's how i chose to write it, with the help of Brian M. Scott's answer.
Let $a$ be an antenna sample, where $a^{cid}$ indicates the cellID and $a^{lac}$ the location area code (lac) of the antenna sample.
Let $T = \{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, ... ,a_{n}\}$ be a logtrace containing a finite sequence of antenna samples $a$ submitted within a time window, and $T^{cid}$ the finite sequence of cellIDs $a^{cid} \in T$. Then, 
Number of distinct cellIDs $=|C|$.
Where $C$ is a set of cellIDs $c \in T^{cid}$.
 A: Your $L$ isn’t actually a set: since it’s intrinsically ordered, it’s a finite sequence, or an $n$-tuple. If you throw away the temporal order, what you have left is a multiset. And if you throw away the information about how many times each cellID appears, you have a set. If $A$ is considered as a multiset, its cardinality is $6$. If, however, it’s considered as a set, which is how you wrote it, then it’s simply equal to $\{1,3,4,5\}$ and has cardinality $4$.
I’m not sure how to answer your notational question, because the answer depends on whether you’re willing to replace $L$ by an intermediate entity first. Technically, $L$ can be viewed as a function from $\{1,\dots,n\}$ to the set of cellIDs. From that point of view the number that you want is $|\operatorname{ran}L|$, the cardinality of the range of $L$. But I suspect that you’d find that a bit clumsy, and it might be best simply to define your own notation, e.g., $n_C(L)$, for the number of distinct cellIDs.
Added in response to edit: @ndrizza: Technically what you’ve written isn’t right, because the terms of a sequence aren’t actually elements of a sequence, and therefore it’s not actually correct to write $c\in T^{cid}$ or $a_t^{cid}\in T$. On the other hand, your meaning is pretty clear, and I’m a bit fussier than most about notation, so it’s entirely possible that it would be acceptable for your intended audience. I might say something like this:

Let $a$ be an antenna sample, where $a^{cid}$ indicates the cellID and $a^{lac}$ the location area code (lac) of the antenna sample.
Let $T = \{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, ... ,a_{n}\}$ be a logtrace containing a finite sequence of antenna samples $a$ submitted within a time window, and $T^{cid}$ be the corresponding finite sequence of cellIDs $a^{cid}$. Let $C_T$ be the set of distinct cellIDs in $T^{cid}$; then $|C_T|$ is the number of distinct cellIDs occurring in $T$.

A: From a formal point of view, as sets $\{1,1,2\}=\{1,2\}$, since $A=B$ means $x\in A\iff x\in B$. You can simply use $|A|$ to denote the number of (distinct) elements in a finite set $A$. Hence $|\{1,1,2\}|=|\{1,2\}|=2$.
