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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 $a \notin S_2$

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  • $\begingroup$ You define $a = (a_1, a_2)$ but that information seems to be irrelevant given the rest of the problem? $\endgroup$ – benguin Feb 28 '17 at 11:56
  • $\begingroup$ If it's of any help, we say that for any sets $S_1$ and $S_2$, that $a \in S_1 \cup S_2$ if and only if $a \in S_1$ and $a \in S_2$. We also say that $a \in S_1 \setminus S_2$ if and only if $a \in S_1$ and $a \notin S_2$. $\endgroup$ – benguin Feb 28 '17 at 11:58
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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 $a_i \in S_i$.

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