Is {{∅}} ⊂ {{∅},{∅}} true Is {{∅}} ⊂ {{∅},{∅}} true or false. I can't decide if this question is true or false. It seems to be false as the sets would be equal? is that correct since an proper subset isn't equal. 
the ⊂ in this means proper subset, the answer is false thanks to the below response. 
 A: Both sets are the same.
The only element in $\{\{\emptyset\}\}$ is $\{\emptyset\}$, and the elements in $\{\{\emptyset\},\{\emptyset\}\}$ are $\{\emptyset\}$ and $\{\emptyset\}$, thus it has only one element, $\{\emptyset\}$.
If $\subset$ denotes proper and not equal subset, it is false.
If $\subset$ denotes proper or equal subset, it is true.
A: If those are the right number of brackets, and that is the proper subset symbol, then your reasoning is okay.   Any set is not a proper subset of itself, and sets do not include redundant copies of their elements.   Thus the LHS is equal to the RHS rather than a proper subset.
Of course, its a different matter if $\bbox[cornsilk,2pt]\subset$ is read as the "subset or equal", which some texts do.
Many consider $\bbox[cornsilk,2pt]\subset$ and $\bbox[cornsilk,2pt]\subseteq$ to be analogous to $\bbox[cornsilk,2pt]<$ and $\bbox[cornsilk,2pt]\leq$ , respectively.   Others just don't make the distinction, causing much confusion to their poor students.
Some authors try to use $\bbox[cornsilk,2pt]\subsetneq$ for proper subset to be clear what they mean; similar to the rarely used $\bbox[cornsilk,2pt]\lneq$ .  However, the tiny strikethrough can be easy to miss if you are not looking out for it.
