# Is $\{\varepsilon\}$ a conventional way to mark the empty language?

I am grading an exercise in Automata and Formal Languages and see many of the students use $$\{\varepsilon\}$$ as the empty language. At first I thought this was an error, and I have asked the lecturer who confirmed. But I keep seeing this and I'm losing my confidence. Is this kind of thing used by anyone? Meaning, does anyone use $$\{\varepsilon\}$$ as the empty language (even though it would "override" the obvious meaning of the singleton of $$\varepsilon$$)?

• It is hard to say if a notation is not being used by anyone. You are a witness that someone is using it, your students. On the other hand, the notation $\{\epsilon\}$ is used in set theory to denote the set with one element $\epsilon$. – plop Mar 26 at 13:34
• You don't have a problem with notation, but rather a problem with students' undestanding, namely the difference between "empty language $\{\}$" and "the language containing just the empty word $\{\epsilon\}$". – Andrej Bauer Mar 26 at 14:01
• It's a common misunderstanding. – Yuval Filmus Apr 1 at 8:50
• There’s also a special notation (which I can’t quite remember) for the regular expression that produces the empty language - the regular language eps produces the regular language consisting of just the empty string. – gnasher729 Apr 1 at 14:46

No, that's the language containing one string: the empty string, which is very much distinct from the empty language. The finite state machine on the left accepts language $$\{\}$$, the finite state machine on the right accepts language $$\{\varepsilon\}$$. They are not the same. As an example, we could use the language as a password check. Anyone submitting an accepted word would be let through. Then $$\{\}$$ means "let no one in" and $$\{\varepsilon\}$$ means "let anyone that submits a blank password in". Those are quite different indeed!