How many maps...
Solution.
We have answers here, so let’s confirm the results of the generalization from Article 1:
| # | \(||A||\) | \(||B||\) | Expression | Result | Correct |
|---|---|---|---|---|---|
| 1 | 4 | 1 | \(1^4\) | 1 | YES |
| 2 | 1 | 4 | \(4^1\) | 4 | YES |
| 3 | 0 | 4 | \(4^0\) | 1 | YES |
| 4 | 4 | 0 | \(0^4\) | 0 | YES |
| 5 | 0 | 0 | \(0^0\) | "it depends" | "mostly" |
The generalization seems to hold up, provided we define our exponential operator in such a way that \(0^0=1\text{.}\) This is a common practice, but I’ve also seen situations where \(0^0\) is left undefined instead. In this context, the answer of 1 to problem 5 makes more sense than than the alternative. In programming terms, it seems perfectly reasonable to think of "no op" as a "function" so there should be an analogous "null map" that maps from nothing to nothing.
