Discrete Math, Fifth Problem Set (June 27)
REU 2003

Instructor: Laszlo Babai
Scribe: Ivona Bezakova

Question: How should we define $0^0$?

Trouble: two conventions conflict: $x^0=1$ and $0^x=0$. We shall argue that this conflict can be resolved and that $0^0=1$ is the reasonable choice.

Argument 1: empty products. The $0^0=1$ convention is consistent with the conventions $\sum_{i\in\emptyset}a_i = 0$ and $\prod_{i\in\emptyset}a_i =1$. (Why are these the only reasonable interpretations of empty sums and products?) The ``empty product $=1$'' rule is used in conventions like $0!=1$ and $a^0=1$.

Argument 2: combinatorial interpretation of powers. Let $A$ and $B$ be two finite sets. The number of functions $f:B\to A$ is clearly $a^b$. Therefore, $0^0=1$ (there is only one function $f:\emptyset\to\emptyset$, namely the empty function).

Argument 3: the limit of $x^y$.

Let us consider $\lim_{x,y\to 0^+}x^y$. This limit does not exist. In fact, subsequences can converge to any number between 0 and 1.

Nonetheless, the limit is ``almost well defined.''