In a related question, an answerer says:
an empty bag is a bag with nothing inside it.
Makes sense, but I'm reading a textbook right now that says:
The empty set has only one subset (namely, itself).
Which seems odd to me. If a set is empty, why would it contain a subset? Regardless of whether or not the subset is also an empty set, the "empty set" still contains a subset, so how is it really "empty"?
To go back to the empty bag analogy, it just sounds like my text book is saying:
an empty bag is a bag with nothing but an empty bag inside it.
Which doesn't make sense because then the bag isn't empty. I can't wrap my head around this concept.
Best Answer
The use of the word "contains" is a bit misleading, there. When we talk about a set $A$ "containing" a subset $B$, what we really mean is that $A$ contains all elements of the set $B$ (that is, for every $x\in B,$ we have $x\in A$). In that sense, we're saying that an empty bag is a bag that contains exactly what an empty bag contains: nothing at all.
It is possible (though not in the case of the empty set) that a set $A$ contains a set $B$ both as an element and as a subset. For example, the set $$A=\bigl\{1,\{1\}\bigl\}$$ has $$B=\{1\}$$ as an element (obvious, hopefully) and as a subset (because $A$ contains every element of $B$). Here's (one place) where the grocery bag analogy breaks down, however, since the grocery bag analogy would suggest that the above $B$ is a subset of the above $A$ by virtue of being contained in $A$ as an element. This is not so. Indeed, if we consider $$C=\bigl\{\{1\}\bigl\},$$ then we find that $B$ is an element of $C,$ but not a subset of $C,$ since $1$ is not an element of $C$!
So, given two arbitrary sets $A$ and $B$, $B$ may be: