# Why does the concatenation of the empty set with any language give the empty set? [duplicate]

Why does the concatenation of $\emptyset$ with any language give $\emptyset$. I would like to know the intuitive explanation for it.

• Just apply the definition. – Raphael Dec 11 '15 at 15:14

Let $L_1, L_2$ be languages, then the concatenation $L_1\circ L_2=\{w\mid w=xy, x\in L_1, y\in L_2\}$. If $L_2=\varnothing$, then there is no string $y\in L_2$ and so there is no possible $w$ such that $w=xy$. Thus for any $L_1$, we'll have $L_1\circ\varnothing = \varnothing$.
• ...because the concatenation of strings is a binary operator, and if $L_{2}$ is empty, then $y$ will not be the empty string, but have no value at all (for the same reason, you can't compute 2 + _). – André Souza Lemos Dec 11 '15 at 15:41