# Notation of intersection between a tuple and a set

Let $a=(a_1,a_2,\dots,a_n)\in [0,1]^n$, an let $B\subseteq [0,1]$. Is it common to denote the set $$\{a_i\mid a_i\in a, a_i\in B\}$$

by $a\cap B$ ? Is there another notation?

Thanks!

Not really no. I think one would understand what you mean by $a\cap B$, but I think it should not be used.

The difference between a tuple and a set is that the tuple is ordered, but the set is not.

It means that the set

$$\{1,2,3\}$$

is equal to the set

$$\{1,3,2,2,1\}.$$

So to avoid confusions like

$$(1,2,1)\cap\{1,1,1\}={?}$$

you should not use $a\cap B$ but your other notation instead.

I would specifically say that using $a\cap B$ to describe the set $$\{a_i\mid a_i\in a, a_i\in B\}$$

is an unwarranted abuse of notation that can only ever confuse the reader. I strongly advise against it.

One interpretation of an $n$-tuple is a function with domain $\{1, \ldots, n\}$; based on that, you could write $\mathrm{ran}\ a \cap B$. Uncouth, perhaps, but not abusive.