Chains and antichains in a family of sets Let $S$ be a family of sets with inclusion as partial order.
Suppose $S$ contains the empty set. Is $\{\emptyset\}$ an antichain?
Also, take a non-empty set $X$ in $S$. Is $\{X\}$ a chain (and/or) an antichain?
Since the question seems to be a bit unclear, let me clarify it with an example.
Let $F=\{\emptyset, \{1,2,3\}\}$ and consider $F$ as a poset ordered by inclusion. My definition of a (anti)chain $A$ is that it is a SET of elements of a poset such that every element in $A$ is (in)comparable. Now $\{1,2,3\}$ is not a chain of $F$ (it is not a set containing some elements of $F$), but is $\{\{1,2,3\}\}$ a chain/antichain?
Is $\{\emptyset\}$ an antichain?

A one-element subset of a poset is always both a chain and an antichain: Any two distinct elements are both comparable and uncomparable.
