Subset of a family of sets I have been struggling to understand what makes a subset of a family of sets. 

Definition: A set $A$ is a subset of another set $B$ if every element of the set $A$ is also in $B$. 

For a family of sets $B$, are the elements of B referring to the sets which $B$ contains, or the elements inside each individual set. For example, say we are considering the  family of sets $$A = \{\{1,2\}, \{3, 4\}, \{5, 6\}\}$$
If F = {3,5}, then is $F$ a subset of $A$?
The whole set $F$ is not in $A$, but the elements of $F$ are.
 A: Question What are the elements of $A$? 
Answer The sets $\{1,2\}, \{3,4\}, \{5,6\}$.
Second Question What are the elements of $F$? 
Answer The numbers $3, 5$.
Are all of the numbers $3,5$ part of $\{1,2\}, \{3,4\}, \{5,6\}$? The answer is no (none of those by the way). So $F$ is not a subset of $A$.
What you can say is that the elements of $F$ are elements of the elements of $A$.
A: A subset of a family of sets is just a (smaller) family of sets, each of which is in the original family.
Using your example, if the family $\mathscr A=\{\{1,2\}, \{3,4\}, \{5,6\}\}$ (which is a family of three sets), then $F=\{3,5\}$ is not a subset of $\mathscr A$. Nor is it an element of $\mathscr A$, for that matter. It is really unrelated to $\mathscr A$.
Continuing the example, you can actually list all of the subsets of $\mathscr A$, each of which is a family of sets:
$$\mathscr B_1=\{\}=\varnothing$$
$$\mathscr B_2=\{\{1,2\}\}$$
$$\mathscr B_3=\{\{3,4\}\}$$
$$\mathscr B_4=\{\{5,6\}\}$$
$$\mathscr B_5=\{\{1,2\}, \{3,4\}\}$$
$$\mathscr B_6=\{\{1,2\}, \{5,6\}\}$$
$$\mathscr B_7=\{\{3,4\}, \{5,6\}\}$$
$$\mathscr B_8=\{\{1,2\},\{3,4\}, \{5,6\}\}=\mathscr A$$
