Is {{∅}} âٹ‚ {{∅},{∅}} true

Is {{∅}} âٹ‚ {{∅},{∅}} 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 âٹ‚ in this means proper subset, the answer is false thanks to the below response.




It is always the case that {x}={x,x}\{x \} = \{x, x\}.
– copper.hat
Oct 21 at 3:09



Some authors use the symbol ⊂\subset to mean “subset”, others use ⊂\subset to mean “proper subset” and ⊆\subseteq to mean “subset”. Please check the definition of ⊂\subset in your book.
– bof
Oct 21 at 3:11


2 Answers


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.



So I think by your explanation it is false. as the âٹ‚ means that the left is in but not the same as the right. correct?
– Collin McCabe
Oct 21 at 3:08



It’s not completely clear but it looks like the OP is using the symbol ⊂\subset to mean “proper subset”. From the question, he seems to know that “both sets are the same”.
– bof
Oct 21 at 3:09



In some books they use ⊂\subset to denote a proper subset, and in others they use that to denote proper or equal subset. In any case, the sets are the same. It depends on what definition you use.
– Antioquia3943
Oct 21 at 3:21

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]<\bbox[cornsilk,2pt]< and \bbox[cornsilk,2pt]\leq\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\bbox[cornsilk,2pt]\subsetneq for proper subset to be clear what they mean; similar to the rarely used \bbox[cornsilk,2pt]\lneq\bbox[cornsilk,2pt]\lneq . However, the tiny strikethrough can be easy to miss if you are not looking out for it.