# What is the difference between language of empty string and empty set language?

I am reading Introduction to Automata theory by Ullman. It says the empty set and set containing empty string is different. I am unable to understand the difference between them as the empty string doesn't have any character. Forgive me if this is dumb but I can't seem to understand it

• $\emptyset$: it's a language containing no string: no element. $\{\epsilon\}$: it's a language containing exactly one string, which has length 0 (the empty string, or "" if you will): one element. – Clement C. Jan 20 '15 at 14:49

The situation is the same with the emptyset and the number $0$ (zero).

$0$ is a number and the set $\{ 0 \}$ has one member, while the emptyset has no members at all.

Numbers are used to perform operations like "counting" objects; in order to correctly describe these operations, it is useful to introduce a number ($0$) that counts no objects.

In the same way, in order to describe the operations performed on strings, it is useful to introduce the string ($\epsilon$) with no symbols (that has $lenght=0$).

• I get it. The empty set doesn't have an element but it differs from set containing empty string in that string operations can be performed on it. – SphericalCow Jan 20 '15 at 15:11
• @user3688659 - exactly; the theory of formal languages is not dealing with sets, but with strings; set language is used (in the meta-theory) to describe the "mechanism" of formal languages. This "mechanism" needs the empty string, which is a string with $0$ lenght. – Mauro ALLEGRANZA Jan 20 '15 at 15:19

Here is a program that accepts the language that contains only the empty string:

    s = get_input();
if (s == "") then ACCEPT;
else REJECT;


Here is a program that accepts the empty language:

    s = get_input();
REJECT;


They are not the same program! The first one accepts the empty string; the second does not.

The set containing one empty string has one element. The empty set has zero elements. The one with one element is "bigger" (its cardinality is larger).