$x \notin \bigcap\limits\_{i\in I}A\_{i}\diagdown \bigcup\limits\_{j\in J}B\_{j} $

This was a problem on exam and I think the correct answer should be 
$x$ such that $x \notin A\_i$ for some $i \in I$ or $x \in B\_j$ for some $j \in J$

The only answer close to this on the test(multiple choice) was $x \notin A\_i$ for some $i \in I$ and $x \in B\_j$ for some $j \in J$

Other answers were:

$x \notin A\_i$ for some $i \in I$ and $x \in B\_j$ for all $j \in J$

$x \notin A\_i$ for all $i \in I$ and $x \in B\_j$ for some $j \in J$

$x \notin A\_i$ for all $i \in I$ and $x \in B\_j$ for all $j \in J$

Which one is correct(if any!) also is there a different way to express the first statement using and?