Is {{∅}} ⊂ {{∅},{∅}} true

Question

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.

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 '16 at 03:11

positron0802 · Accepted Answer · 2016-10-21T03:21:23.083

2

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.

edited Oct 21 '16 at 03:21

answered Oct 21 '16 at 03:04

positron0802

5,652

So I think by your explanation it is false. as the ⊂ means that the left is in but not the same as the right. correct? – Collin McCabe Oct 21 '16 at 03:08
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 '16 at 03:09
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. – positron0802 Oct 21 '16 at 03:21

score 0 · Answer 2 · answered Oct 21 '16 at 03:54

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.

Is {{∅}} ⊂ {{∅},{∅}} true

2 Answers2