What is the best approach to checking the validity of a set given the description below?

My knowledge of Sets is not be best so bare with me and I will appreciate corrections to any mistakes noticed :).

I was thinking through an algorithm to check the validity of a kind of set.

Description:

Given the sets $$A_1$$, $$A_2$$, $$A_3$$, $$A_4$$, $$A_5$$,

And a $$Set A$$ such that $$A = \{A_1, A_2, A_3, A_4, A_5\}$$

I consider an arbitrarily provided set $$S_{input}$$ (input to the algorithm) to be valid if for every set that is an element of $$A$$ (call it $$A_i$$), the intersection of $$S_{input}$$ and $$A_i$$ is not empty. That is: $$A_i \cap S_{input} \neq \emptyset$$ .

In other words:

In my algorithm an arbitrarily provided set $$S_{input}$$ is considered to be a valid set ($$S_{valid}$$), if and only if $$S_{input}$$ is a set such that $$\forall A_i \in A; A_i \cap S_{input} \neq \emptyset$$ where $$A$$ is a previously known set of sets.

Some examples:

If $$A = \{ \ \{1,2,3,4\},\ \{1,2\},\ \{3\}, \ \{1, 4\}, \ \{1,2,3,4,5,6\} \ \}$$

1. $$S_{input} = \{1,2,3\}$$ is Considered a valid set $$S_{valid}$$ because for every set $$A_i$$ in $$A$$, you get the following results:

• $$\{1,2,3,4\} \cap \{1,2,3\} = \{1,2,3\} \neq \emptyset$$
• $$\{1,2 \} \cap \{1,2,3\} = \{1,2\} \neq \emptyset$$
• $$\{3\} \cap \{1,2,3\} = \{3\} \neq \emptyset$$
• $$\{1,4\} \cap \{1,2,3\} = \{1\} \neq \emptyset$$
• $$\{1,2,3,4,5,6\} \cap \{1,2,3\} = \{1,2,3\} \neq \emptyset$$
2. $$S_{input} = \{1,2,4\}$$ is Considered an invalid set because for every set $$A_i$$ in $$A$$, you get the following results:

• $$\{1,2,3,4\} \cap \{1,2,4\} = \{1,2,4\} \neq \emptyset$$
• $$\{1,2 \} \cap \{1,2,4\} = \{1,2\} \neq \emptyset$$
• $$\{3\} \cap \{1,2,4\} = \{\} = \emptyset$$
• $$\{1,4\} \cap \{1,2,4\} = \{1, 4\} \neq \emptyset$$
• $$\{1,2,3,4,5,6\} \cap \{1,2,4\} = \{1,2,4\} \neq \emptyset$$

Since the third set ($$\{3\}$$) results in an empty set when intersected with $$S_{input}$$, $$S_{input}$$ is not valid.

My questions:

1. Is this a known type of set operation? Basically is there a name for such a set $$S_{valid}$$ (based on my definition above) that I don't know of? (by this definition there should be may such sets $$S_{valid}$$ for any given $$A$$)

2. Will there exist a set (call it $$A_u$$) such that for a given set $$S_{input}$$, if all the elements of $$S_{input}$$ are also members of $$A_u$$ then $$S_{input}$$ can be considered to be a valid set ($$S_{valid}$$) based on my above conditions? In other words, does there exist $$A_u$$ such that if $$\forall e \in S_{input}, e \in A_u$$ then $$S_{input}$$ is considered a valid set $$S_{valid}$$ ?

3. Any efficient suggestions (preferable pseudocode) on a routine to check if a given set $$S_{input}$$ is a valid set $$S_{valid}$$?

4. Any suggestions on (preferably pseudocode) on a routine to generate a set $$A_u$$ described in question 2?

1. The closest term might be "pairwise non-disjoint", but that's a stretch. This problem reminds me a bit of the sunflower lemma, but almost reversed: a set $$W$$ is called a sunflower if no matter which $$w,v\in W$$ we choose, $$w\cap v$$ is always the same set, called the "kernel" of $$W$$. For the sunflower problem, it doesn't matter if the sets $$w\cap v$$ are empty, as long as they are equal for any choice of $$w,v$$. In your problem, it doesn't matter if the sets $$w\cap S_{input}$$ are unequal, as long as they aren't empty for any choice of $$w$$.
2. No. Assume there was such a set $$A_u$$. Then if we set $$A=\{A_u,\{A_u\}\}$$, $$S_{input}=\{A_u,\{A_u\}\}$$ is a counterexample, in particular $$A_u\cap\{A_u\}$$ is empty by the axiom of regularity.
3. It's not really possible to compute because if $$A$$ is infinite, there will be infinitely many intersections to check. You can think of it as a "foreach" instruction, where "$$S_{input}$$ is valid" is the Boolean and applied to all statements "$$A_i\cap S_{input}\neq\varnothing$$" for $$A_i\in A$$