Proper notation of a condition

Question

Let's say that we have two different sets, $S_1,S_2$, and also that you have a combination $a=(a_1,a_2)$

Which is the proper way to notate the following:

a) something holds if $a \in S_1$ or $a \in S_2$,

b) something holds if $a \in S_1$ and at the same time $a \notin S_2$

Answer 1

You could write

1. $a \in S_1 \cup S_2 \Rightarrow \dots$
2. $a \in S_1 \setminus S_2 \Rightarrow \dots$

although I suspect you want to express some logical combination of the conditions $a_i \in S_i$.