Total order of given set

Question

For a given set $A$, if cardinality of $A$ is equal to zero or $1$, can we conclude that $\left \langle \mathcal{P}(A),\subseteq \right \rangle$ is a total ordering? I think when cardinality is one, then empty set and set $A$ will be comparable, but what about an empty set?

Answer 1

Well, for the case $A = \varnothing$, then $\mathcal P(A) = \{\varnothing\}$.

Take $S,T \in \mathcal P(A)$. Since $|\mathcal {P}(A)| = 1$ necessarily, it follows that $S = T = \varnothing$. (The sets need not be distinct.) The empty set is a subset of every set, so you will have both $S \subseteq T$ and $T \subseteq S$ (as $\varnothing \subseteq \varnothing$).

Thus, you have comparability (or connexity, as it is sometimes called).