Understanding subset terminology I'm having a hard time wrapping my head around subsets/element problems.
For example:
Is $\{2\} \subseteq \{2,3\}$?
I thought it was because every element in the left set is also in the set on the right.
Furthermore is
$\{2\} \subseteq \mathcal{P}(\{1,2\})$?
I also thought this was true because the set on the left contains one or more of the elements of the set on the right.
 A: Suppose $A$ is a set. Every element that appears in $A$ is an element of $A$.
So for the first example, consider the set $A = \{2,3\}$. We see that $2 \in A$.
By definition, $S$ is a subset of $A$ ($S \subseteq A$) if and only if every element of $S$ is in $A$.
Therefore, as you said, $\{2\} \subseteq A.$
For the second example, recall the definition of power-set.
Let $A$ be any set. The power-set of $A$ is the set consisting of all the subsets of $A$.
Considering a set $B = \{1,2\}$, we get that
$$\mathcal{P}(B) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}.$$
We see that $\{2\} \in \mathcal{P}(B)$ but $\{2\} \not \subseteq \mathcal{P}(B)$. But it is true that $\{\{2\}\} \subseteq \mathcal{P}(B)$, given our definition of subset [Every element in $\{\{2\}\}$ (in this case, it is only $\{2\}$) is an element of $\mathcal{P}(B)$],
A: $\{2\} ⊆ \{2,3\}$ because the set $\{2\}$ has elements all contained in set $\{2,3\}$.
While $\{2\} ⊆ P(\{1,2\}) = \{Ø ,\{1\},\{2\},\{1,2\}\}$ is false because there are elements of set $\{2\}$, namely $2$, which aren't contained in $P(\{1,2\})$ : It contains just other sets, not numbers.
