Is Your Set Builder Notation Correct for Describing Pairs in Sets S and W?

[ektrules]

Ok, I'm not very familiar with set notation. I was just wondering if the following is correct notation and means what I think:

{x$\in$S : $\exists$y$\in$x, y$\in$W}

S is a set of pairs of symbols (tuples of length 2 is the technical term I believe). W is a set of symbols.

What I want is the set of pairs in S that contain at least 1 symbol from set W.

Does my set builder notation correctly describe what I'm looking for? I don't necessarily have to use set builder notation; I just can't think of a way to describe it with unions, intersections, and quantifiers.

 

[CompuChip]

I think it is correct, however it may become slightly clearer if you explicitly write down the pairs:

$$\{ (x, y) \in S \mid x \in W \vee y \in W \}$$

If you would like to omit the set builder notation, you could consider something like
$$(W \times X) \cup (X \times W)$$
where X is the set of all symbols (and $W \subseteq X$).