How is an empty set truly "empty"?

Solution 1:

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:

• a subset of $A$ but not an element (e.g.: for any set $A$ such that $\emptyset\notin A$, let $B=\emptyset$),
• an element of $A$ but not a subset,
• an element and a subset of $A$,
• neither an element nor a subset of $A$ (consider $A=\{1\},$ $B=\{3\}$).

Solution 2:

Do you know what it really means for one set to be a subset of another set? If the set $A$ is a subset of the set $B$, then we write $A\subseteq B$; now, to show that $A\subseteq B$, we must show that $x\in A\to x\in B$.

The fact that $\varnothing\subseteq\varnothing$ is really not that surprising if you are aware of what a so-called "vacuous truth" is. That is, $x\in\varnothing$ is clearly not true; thus, we can freely conclude (i.e., "vacuously") whatever we want. Hence, $x\in\varnothing\to x\in A$, where $A$ is any set whatever, including the empty set.

Does that make things any clearer? It seems like you want a materialistic or intuitive way of seeing that $\varnothing\subseteq\varnothing$ when really the key lies in understanding what it means for one set to be the subset of another set and also what a vacuous truth is and how you can use that.