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 $\varepsilon$)?

  • $\begingroup$ 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 $\epsilon$. $\endgroup$ – plop Mar 26 at 13:34
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    $\begingroup$ 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 $\{\epsilon\}$". $\endgroup$ – Andrej Bauer Mar 26 at 14:01
  • $\begingroup$ It's a common misunderstanding. $\endgroup$ – Yuval Filmus Apr 1 at 8:50
  • $\begingroup$ 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. $\endgroup$ – 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.


If a lot of students are confused maybe explicitly show the difference in an exercise/during a lecture.

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!

  • $\begingroup$ Thanks. I am just grading the exercise, I am not the lecturer. Just needed to know that I was not wrong. $\endgroup$ – Yekhezkel Yovel Mar 26 at 18:00

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