Mathematical Properties of awareness

Awareness has the following properties:
1. Awareness is reflexive. I’m aware of myself.
2. Awareness is not symmetric. I know you. It does not imply you know me.
3. Awareness is not transitive. A knows B. B knows C. It does not imply A knows C.
Note: A family, on the other had, is reflexive, symmetric, and transitive.
4. Being known to everyone is set specific.
5. It is possible that a set contains a celebrity who is known to everyone in the set. But in a city not everyone knows each other.
6. There does not exist a universal celebrity who is well-known globally. Only a large fraction of people may know her.
7. Knowledge may be language invariant. But due to the language barrier, it cannot be fully transformed.