Set notation and the difference between $\subseteq,\in,\subset$. 
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*What does it mean to say $\mathcal F$ is a family of subsets? (an example would be much appreciated :)) What would be a layman's example?      

*When $B=\{b,c\}$ is it appropriate to write $\{a,\{b,c\}\} \subseteq A \Rightarrow\ B\subseteq A$

*$\{a, \{b,c\} \} \in A$ is that good notation? 

*Would $\{\{a,d\}, \{b,c\}\}$ be considered as a family of subsets?  

 A: *

*Just that $\mathcal F$ is a set of subsets, of some stated set. "Family" is a more informal term when it helps to think of sets of sets as different from elements (in set theory, this is a relative term only).

*No. Just do the substitution. $\{a,B\} \subset A$ means $B \in A$. Just like $\{1,2\} \subset \mathbb N$ means $2 \in \mathbb N$.

*Yes, but only if you are intending to write something really complex: "The set that has two elements, the first of which is $a$ and the second of which is the set $\{b,c\}$ itself, itself, is an element of $A$.

*I guess it's a family of subsets of $\{a,b,c,d\}$. Again, "family" is a relatively informal term usually, so your (4) would only really make sense if you told me about $\{a,b,c,d\}$ first, and asked me if your example were a family of subsets of that four-element set.


To answer your title question, $\subset$ and $\subseteq$ depend on the author's choice of convention. Usually they mean the same thing, but sometimes an author wants $\subset$ to mean "is a strict subset", that is, not the whole set itself, and $\subseteq$ to mean "is a subset or equal." Presumably this follows how $<$ and $\leq$ work. However, in the usual practice, $\subset$ means "is a subset (or equal)", $\subseteq$ means the same thing, and $\subsetneq$ means "is a strict subset of".
A: 1) $\mathcal{F}$ is a family of subsets if it's elements are sets. Good example would be power set of given set. Another example : consider rolling a dice - the set of possible outcomes will be $\{ 1, 2, 3, 4, 5, 6 \}$ here $\mathcal{F}$ could be $\{ \{2, 4, 6\}, \{1, 3, 5\}\}$  
2) No. Here $A$ and $B$ are to be understood as collections of sets
3) Yes - if $A$ is collection of sets
4) Yes - by definition!
A: *

*A collection $\mathcal F$ of of a given set $S$ is called family of subsets of $S$.


Example. Let $S=\{a,b,c,d,e\}$, example of family of subsets would be $\mathcal F=\{A_1,A_2\}$  where $A_1=\{a,b,c\},A_2=\{d,e\}.$


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*No, you can say that $B \in A$ though.

*Yes. Here $A$ is collection of sets.

*Yes,
