What does "$A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of $\mathcal{P}(N)$" mean?

Question

I have read the following in some exercise for discrete mathematics. Let $N$ be a set and $\mathcal{P}(N)$ be its power set. Then $A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of $\mathcal{P}(N)$.

I am not really sure what is asked here. I know how to show that $\subseteq$ is a partial order on the power set but I am not sure what exactly is asked here. What order relation do I need to show and what should I do with $A \leq B$?

Answer 1

A binary relation $\leq$ over a set $X$ is an order if the following hold:

1. $\leq$ is reflexive, i.e. $n \leq n$ for all $n \in X$;
2. $\leq$ is antisymmetric, i.e. if $n \leq m$ and $m \leq n$ then $n = m$, for all $n,m \in X$;
3. $\leq$ is transitive, i.e. if $n \leq m$ and $m \leq \ell$ then $n \leq \ell$, for all $n, m, \ell \in X$.

Often an order relation on $X$ is called partial to distinguish it from a total order, which is an order on $X$ such that, for all $n,m \in X$, one has $n \leq m$ or $m \leq n$.

You are asked to prove that $\subseteq$ is a reflexive, antisymmetric and transitive relation on $\mathcal{P}(N)$.

Answer 2

A "ordered set" is a shortcut for "partially ordered set", so you must prove the 3 axioms of a ordered relation. See here.