# In formal languages, why is $L^0 = \{ \epsilon \}$? Why isn't it the empty set ∅?

It just doesn't make logical sense to me that a language to the power of $1$, is itself, but to the power of $0$ is only a tiny part of itself

• Wouldn't it would make much more sense if $L^0 = ∅$.

In English that would say, if you have this language in zero magnitude, you've got absolutely nothing.

Also, a related question I have is,

• Why is $∅* = \{ \epsilon \}$? How does concatenating the empty set with itself infinitely many times magically create a non-empty set?
• Does an analogy with $x^0 = 1$ (for $x \neq 0$) not help? – James May 27 '15 at 15:26
• Not every language contains $\epsilon$. – mrp May 27 '15 at 15:31

## 1 Answer

You want $L^nL = L^{n+1}$ (law of indices). You also clearly want $L^1 = L$. Combining them gives $L = L L^0$, so $L^0 = \{\epsilon\}$.

By the way, the same sort of reasoning explains why defining $n^0$ to equal $1$ when $n$ is an integer is a good idea.

• $0^0=1$ also, and the analogy for languages is that $\emptyset^0 = \{\epsilon\}$. – MJD May 27 '15 at 15:34