The problem I am working on is, "What is the covering relation of the partial ordering $\{(A,B)|A⊆B\}$ on the power set of $S$, where $S=\{a, b, c\}$?"
I am reading the answer key, and I can follow it for the most part. What I don't understand is this, "In this problem $A \preceq B$ when $A \subseteq B$. For $(A,B)$ to be in the covering relation, we need $A$ to be a proper subset of $B$ but we must also have no subset strictly between $A$ and $B$."
I understand the part about having no subset between $A$ and $B$, that would contradict the very definition of of a covering relation. For example, I know that $(\{a\},\{a,b,c\})$ is not in the covering relation, because of the the ordered-pair $(\{a,b\},\{a,b,c\})$, meaning that $\{a,b\}$ is an intermediate value. The part that I don't understand is why $A$ has to be a proper subset of $B$. Why is that?
Edit: Let $(S,\preceq)$ be a poset. We say that an element$y∈S$ covers an element $x∈S$ if $x≺y$ and there is no element $z∈S$ such that $x≺z≺y$. The set of pairs $(x, y)$ such that $y$ covers$x$ is called the covering relation of $(S,\preceq)$. From the description of the Hasse diagram of a poset, we see that the edges in the Hasse diagram of $(S,\preceq)$ are upwardly pointing edges corresponding to the pairs in the covering relation of(S,). Furthermore, we can recover a poset from its covering relation, because it is the reflexive transitive closure of its covering relation. (Exercise 31 asks for a proof of this fact.) This tells us that we can construct a partial ordering from its Hasse diagram.