Understanding issue of basic sets I have some answers here, that I can barely understand.
1) $\{a,b,\{a,b\}\} - \{a,b\} = \{a,b\}$. The answer indicates it is wrong. I think it is correct, what is it that I cant see? (very confused here).
2) $\{a\} \subseteq \{a,b,\{\{\{a\},b\}\}\}$. Correct. Mr. obvious (?) ...
3) $\{a\} \in \{a,b,\{\{\{a\},b\}\}\}$. Wrong. Why is it wrong??? I thought it was correct. 
Anyone here could perhaps explain these to me?
 A: $(1): \{a, b, \{a, b\}\} - \{a, b\} = \{\{a, b\}\}$
$(2)$ is correct. $$a \in \{a, b, \{\{\{a\},b\}\}\} \implies \{a\}\subseteq \{a, b, \{\{\{a\},b\}\}\}$$
$(3)$: $$a \in \{a, b, \{\{\{a\},b\}\}\} \implies \{a\}\subseteq \{a, b, \{\{\{a\},b\}\}\}$$ but not $$\{a\} \in \{a, b, \{\{\{a\},b\}\}\}$$
A: *

*If $A$ and $B$ are sets, the elements of $A\setminus B$ are the objects that are in $A$ but not in $B$. The elements of $\big\{a,b,\{a,b\}\big\}$ are $a,b$, and $\{a,b\}$, and the elements of $\{a,b\}$ are $a$ and $b$. Thus, the only element of $\big\{a,b,\{a,b\}\big\}$ that is not an element of $\{a,b\}$ is the object $\{a,b\}$, and the set difference $\big\{a,b,\{a,b\}\big\}\setminus\{a,b\}$ is the set whose only member is that object; that set is $\big\{\{a,b\}\big\}$.

*You’re okay here.

*The elements of the set $\big\{a,b,\{\{\{a\},b\}\}\big\}$ are $a$, $b$, and the set $\big\{\{\{a\},b\}\big\}$. None of these is $\{a\}$, so $\{a\}\notin\big\{a,b,\{\{\{a\},b\}\}\big\}$. Remember, $a$ and $\{a\}$ are not the same thing.
For the record, the set $\big\{\{\{a\},b\}\big\}$ has one element, the set $\big\{\{a\},b\big\}$. This set in turn has two members, the set $\{a\}$ and the object $b$.
A: 1) substructing {a, b} means to "delete" a and b, not the set containing them.
3) the group {a} is not member in {a,b,{{{a},b}}} since it not a, b, and not {{{a},b}}. So any member of the set is not {a}
Hope it helps.
A: For $\{a,b,\{a,b\}\}\setminus\{a,b\},$ we are removing the elements $a$ and $b$ from $\{a,b,\{a,b\}\}$, and so the only remaining element is $\{a,b\}.$ Hence, the answer is $\{\{a,b\}\}.$
You are correct on the second one. Every element of $\{a\}$ is an element of $\{a,b,\{\{\{a\},b\}\}\}.$
While $\{a\}$ is a subset of $\{a,b,\{\{\{a\},b\}\}\},$ it is not an element. There are only three elements of the given set: $a,b$ and $\{\{\{a\},b\}\}.$
