# Properties of binary relations

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.

• It helps a little bit. I am just confused on how you knew to introduce another set $U$. How do I go about checking something like this where we just say "Let $A$ be some set." Do I just plug in some values? So like, let $A = \{ a, b, c \}$? Apr 19 '15 at 19:39
• You know, if you don't like the set $U$, just forget about it. I introduced it to be able to speak of the relation $\subset$ as a relation on some set (in my case $U$). If you wish, you can say that $\subset$ is a relation, and it contains a pair of sets $(A,B)$, if $A\subset B$. Oh, and concerning if you should just plug in some values, the answer is "it depends". If you want to prove something, you need to prove it for all sets, not just specific ones. If you want to disprove something, it suffices to take some sets that show that whatever you want to disprove is not true. Apr 19 '15 at 19:43
• I mean the $U$ does help. Apr 19 '15 at 19:46