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Next: Number Theory

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.


\begin{exercise}
% latex2html id marker 826Let $0\le \alpha\le 1$. Prove: ther...
...\lim_{n\to\infty}y_n=0$,
and $\lim_{n\to\infty}x_n^{y_n}=\alpha.$\end{exercise}

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


\begin{exercise}
% latex2html id marker 828Prove that $\lim_{x,y\to 0^+}x^y$\ ...
...ear definition and prove this statement based on your definition.
\end{exercise}





Varsha Dani 2003-07-25