I am so lost on this concept. We are doing some problems over properties of binary sets, so for example: reflexive, symmetric, transitive, irreflexive, antisymmetric.
This particular problem says to write down all the properties that the binary relation has: The subset relation on sets.
I am completely confused on how to even start this. Should I use one set $A$? Or have two sets, $A$ and $B$? So the relation is $A \subset B$, I think? I'm not even sure on that.
So for reflexive we have to prove $xRx$ for all $x \in A$. That's proving that one set is reflexive. I'm not sure what to do with two sets. I mean, I could do $xRx$ for all $x \in A$ and $xRx$ for all $x \in B$, but I'm not sure that even helps me at all since I am trying to find properties for $A \subset B$. Can anyone maybe help explain this concept to me?
To clarify, the relation the question is looking for is the following. Let $U$ be some set, and consider the relation $\subset$ on $U$. What does this relation do? It relates a subset $A\subseteq U$ to another subset $B\subseteq U$, if $A$ is a proper subset of $B$.
So you need to check if this relation ($\subset$) is reflexive, symmetric etc.
For instance, it is not reflexive, because no set $A$ is a proper subset of itself, and it is not symmetric, because if we have $A\subset B$ then we cannot have $B\subset A$.
Notice! It is not clear from the question, if you should consider the subset relation $\subset$ or the subset relation $\subseteq$, so I suggest considering both.
I hope this makes it clearer.