# Why $\phi$ $\cdot$ R = $\phi$, rather than $\phi$ $\cdot$ R = R in Automata? [duplicate]

I understand that $$\phi$$ is a null symbol.
why concatenation of any language L with $$\phi$$ is $$\phi$$ rather than L ?

• Try using the definition of concatenation. – Yuval Filmus Mar 23 '19 at 9:44

For two sets of strings $$S_1$$ and $$S_2$$, the concatenation $$S_1\cdot S_2$$ consists of all strings of the form $$vw$$ where $$v$$ is a string from $$S_1$$ and $$w$$ is a string from $$S_2$$, or formally $$S_1\cdot S_2 = \{ vw : v \in S_1, w \in S_2 \}$$.
What about $$\emptyset\cdot R$$?
Since there is no string in the empty set, we cannot find any string of that form $$vw$$ where $$v$$ is a string from the empty set. For example, you cannot form a mixed double in tennis if there is no male players. So $$\emptyset\cdot R=\emptyset$$.
You're probably confusing $$\emptyset$$, the langauge that contains no strings at all, with $$\{\varepsilon\}$$, the language that contains only the empty string.