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Not a duplicate. This deals with sets that are elements of other sets. In other words, sets of sets. The other question deals with elements of sets and subsets of sets. The key difference in here is that the sets are the elements. Besides the other question only asks if $A \subset A$ which is trivial.

When it comes to sets of sets, $ \subset$ and $\in$ always confuse me in set theory. Could someone explain in detail with an example? I greatly appreciate it.

In other words, when do we say $\mathbb{A} \in \mathbb{B}$ and when do we say $\mathbb{A} \subset \mathbb{B}$ (besides the definition that every element in $\mathbb{A}$ ought to be in $\mathbb{B}$)

Thank you

Bachelier
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    If $A = {a,b,c}$, then the elements of $A$ are $a$, $b$, $c$, while the subsets of $A$ are $\emptyset$, ${a}$, ${b}$, ${c}$, ${a,b}$, ${a,c}$, ${b,c}$, ${a,b,c}$. So, $a \in A$ and ${a} \subset A$, but $a \not\subset A$ (unless $a$ is also a set) and ${a} \not\in A$. Is this the kind of example you are interested in? Is it helpful? – Fabio Somenzi Jan 31 '18 at 18:43
  • So subsets of $A$ contain elements of $A$. – Bachelier Jan 31 '18 at 19:28
  • Not a duplicate. This deals with sets that are elements of other sets. In other words, sets of sets. The other question deals with elements of sets and subsets of sets. The key difference in here is that the sets are the elements. – Bachelier Feb 01 '18 at 01:10

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when do we say $\mathbb{A} \in \mathbb{B}$

When $\mathbb{B}$ is a set and $\mathbb{A}$ is one of its elements. It's would be unusual to style a set and one of its elements in the same way. Still, you can have sets with sets as elements: let $\mathbb A = \{1,2\}$, and $\mathbb B = \{3,\{1,2\}\}$. Then you could say $\mathbb A \in \mathbb B$.

when do we say $\mathbb{A} \subset \mathbb{B}$ (besides the definition that every element in $\mathbb{A}$ ought to be in $\mathbb{B}$)

Well, never, other than the definition. The statement $\mathbb{A} \subset \mathbb{B}$ is defined to mean that every element of $\mathbb A$ is an element of $\mathbb B$. If $\mathbb A$ and $\mathbb B$ are as defined above, then $\mathbb A \not\subset \mathbb B$. If you set $\mathbb B = \{3,1,2\}$ instead, then yes, $\mathbb A \subset \mathbb B$.

Matthew Leingang
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