Why is true that the empty set is a proper subset of {x} but it is not $\in$ x?
Is this just a thing to memorize or is there some fundamental concept at play here? To me it seems like this is just a rule with a wink and a nod, as in it has no real logic to it at this point.
Best Answer
The notation $a\in B$ denotes that the element $a$ is actually a member of the set $B$.
The notation $A\subseteq B$ denotes that every element of $A$ is also an element of $B$.
In your example: every element in the empty set $\{\}$ is certainly also an element of $\{x\}$... this is vacuous, because there ARE no elements in $\{\}$.
On the other hand, $\{\}$ is not an element of $\{x\}$; if it were, your set would look like, for instance, $\{x,\{\}\}$. It would literally mean that the empty set was an element of the set in question... but in this case, that set contains only one element, namely $x$.