Understanding concatenating the empty set to any set.

Solution 1:

It turns out from the comments that the context is regular sets. If $A$ and $B$ are sets, we define $A\circ B=\{ab:a\in A\text{ and }b\in B\}$. If $B=\varnothing$, there are no objects $b\in B$, so there are no objects $ab$ such that $a\in A$ and $b\in B$; thus, $A\circ\varnothing=\varnothing$.

Solution 2:

The wrong line of thinking is almost correct, in the following way. Let $\epsilon$ be the empty string. For any string $a$, we have $a\epsilon=a$. Let $\{\epsilon\}$ be the set containing exactly one element, namely the empty string. Then for any set of strings $A$, we have $A\circ\{\epsilon\}=A$.

The real issue, then, is confusing $\{\epsilon\}$ with $\varnothing$. The former is a set containing one string; the latter is a set containing zero strings.

(One potential trap is that some formalisms might identify $\epsilon=\varnothing$. Then we have to distinguish between $\{\varnothing\}$ and $\varnothing$. This is theoretically straightforward, but it means that you have to be careful with your notation.)