Dating/People and properties/All relation/Exercise

On the dating-site "Catch your match“, there is a set ${}M$ of registered users. Moreover, there is a set ${}E$ of properties, which the users satisfy or not (this can be seen in their profiles). For a subset of properties ${}W\subseteq E$ (wanted properties), we define

${}W^{\perp }={\left\{m\in M\mid m{\text{ fulfills all properties from }}W\right\}}\,,$

and for a subset ${}S\subseteq M$, we define

${}S^{\perp }={\left\{e\in E\mid {\text{ every person from }}S{\text{ fulfills the property }}e\right\}}\,.$

1. Describe, for a property ${}e\in E$, the set ${}\{e\}^{\perp }$ with a sentence.
2. Describe, for a person ${}m\in M$, the set ${}\{m\}^{\perp }$ with a sentence.
3. Why do we (probably) have ${}E^{\perp }=\emptyset$?
4. Show: for subsets ${}W_{1}\subseteq W_{2}$ (in ${}E$), we have

   ${}{\left(W_{1}\right)}^{\perp }\supseteq {\left(W_{2}\right)}^{\perp }\,.$

5. Show: for an arbitrary subset ${}W\subseteq E$, we have

   ${}W\subseteq {\left(W^{\perp }\right)}^{\perp }\,.$

6. Show: for a union

   ${}W=W_{1}\cup W_{2}\subseteq E\,,$

   we have

   ${}{\left(W_{1}\cup W_{2}\right)}^{\perp }={\left(W_{1}\right)}^{\perp }\cap {\left(W_{2}\right)}^{\perp }\,.$

7. Does, for an intersection

   ${}W=W_{1}\cap W_{2}\subseteq E\,,$

   the relation

   ${}{\left(W_{1}\cap W_{2}\right)}^{\perp }={\left(W_{1}\right)}^{\perp }\cup {\left(W_{2}\right)}^{\perp }\,$

   hold?

8. Does, for an arbitrary subset ${}W\subseteq E$, the relation

   ${}W^{\perp }={\left({\left(W^{\perp }\right)}^{\perp }\right)}^{\perp }\,$

   hold?

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