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Suppose there are two sets (spaces) X and Y. Given N subsets of $X \times Y$: $S_1, \dots, S_N \subseteq X \times Y$. I need to compute the following set $S_X \subseteq X$:

$ S_X = \{x \in X : \forall y \in Y, \exists k: (x,y) \in S_k \} $ In words: $S_X$ is the set of $x \in X$ such that for every $y \in Y$, there exists a set $S_k$ (among $S_1,\dots,S_N$) that contains $(x,y)$. Of course, $S_X$ may be empty.

My question is how to compute $S_X$, assuming that all basic set operations are available, e.g. union, intersection, projection?

If it helps then my problem context is more specific: $X = \prod_{i=1}^n[a_i, b_i] \subset \mathbb{R}^n$, $Y = \prod_{i=1}^m [c_i, d_i] \subset \mathbb{R}^m$ and $S_k$ are polytopes in $X \times Y \subset \mathbb{R}^{m+n}$.

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    He means, whether it can be described as a combination of unions, intersections, projections, set differences, etc using the "original" sets.2012-03-17

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$\overline{\pi_X(\overline{\cup S_i})}$

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    Brilliant! Or silly me :-)2012-03-17