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

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Problem Detail:

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

#### Answered By : Rick Decker

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 $L_1\circ\varnothing = \varnothing$.