# Show that the relation on the set of all relations is a partial order

I've tried looking at similar problems but what confuses me here is that this regards relationships between relations (if I've understood it correctly).

My method has been to try and show that $$\leq$$ satisfies the three conditions of a partial order, but how do I actually for instance show that if ~1 $$\leq$$ ~2 and ~2$$\leq$$~1 this implies ~1=~2?

• Conceptually it is the same as the relation $\subseteq$ on $A^2$. So you can follow the same proof as showing $\subseteq$ is a partial order. $R\subseteq S\subseteq R$ implies $R=S$. Commented Feb 11, 2021 at 12:19
• Could you expand a bit on that? Why is it conceptually the same? Commented Feb 11, 2021 at 14:34
• @Chrystomath So do I treat each ~i as a set? And then show that for all ~i$\in R$ the three standard conditions hold as you said? Or work directly with R? Commented Feb 11, 2021 at 15:05
• $\sim$i *is* a subset of $A^2$; it is an element of $\mathcal{R}$. So yes to your questions. Commented Feb 11, 2021 at 15:35