What does "ordering of sets by inclusion" mean? I was recently introduced to partial order relations. Although I understand the concept of a relation, I do not understand how subsets can have a sequence, or what that has to do with the phrase "ordering of sets by inclusion." Can anyone help me clarify the concept with an intuitive example?
 A: Let $\mathcal{S}$ be a collection of sets.  Then for sets $S_1, S_2 \in \mathcal{S}$, we can define $S_1 \prec S_2$ to mean $S_1 \subset S_2$.
So for example, considering sets of integers, we can say $\{1, 2, 3\} \prec \{1, 2, 3, 4\}$.  This is a partial order because not all sets are comparable.  For instance $\{1, 2, 3\}$ and $\{1, 3, 5\}$ are not comparable because neither is a subset of the other.
A: Suppose you have a set $\mathtt{S}$.  Now denote power set of $\mathtt{S}$ by $\mathcal{P}(\mathtt{S})$. Then $\mathcal{P}(\mathtt{S})$ is a poset under the relation $\subseteq$ (inclusion).


*

*Reflexive- As $A \subseteq A$ for all $A \in \mathcal{P}(\mathtt{S})$

*AntiSymmetric- As $A\subseteq B$ and $B \subseteq A$  $\implies A=B$

*Transitive- You can see it now, I am sure.


So it is a partial order on $\mathcal{P}(\mathtt{S})$. 
On any set of sets, inclusion is a partial order.
A: Given any family of sets $\mathcal{F}$ there is a poset $P=(\mathcal{F},\{(A,B)\in\mathcal{F}^2:A\subseteq B\})$ corresponding to that family ordered by inclusion. Now by an "inclusion maximal/maximum/minimal/minimum" set in $\mathcal{F}$ what is meant is simply a maximal/maximim/minimal/minimum element of $\mathcal{F}$. Also its worth noting that every partial order is isomorphic to a family of sets ordered by inclusion, in particular the principal lower sets of any partial order when ordered by set inclusion are always isomorphic to said partial order.
