I know that concatenating the empty set to any set yields the empty set. So, $A \circ \varnothing = \varnothing$. Here $A$ is a set of strings and the concatenation ($\circ$) of two sets of strings, $X$ and $Y$ is the set consisting of all strings of the form $xy$ where $x\in X$ and $y \in Y$. (You may want to take a look at page 65, Example 1.53 of Introduction to the Theory of Computation by Michael Sipser).
However, I get somewhat puzzled when I try to intuitively understand this.
A wrong line of thinking will make one to ask, "If we concatenate $A$ with $\varnothing$, should not it still be $A$?"
Well, one way force myself to understand the correct answer, may be, to say that, since I am concatenating with an empty set, actually I will not be able to carry out the concatenation. The concatenation will not exist at all.
I am asking for help from experienced users to provide hints and real life examples which will help one to modify the thinking process and help one better to really understand the correct answer. I am putting more stress on real life examples.
I need to understand this. I am not happy simply memorizing the correct answer.
Best Answer
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, $A\circ\varnothing=\varnothing$.