# Write Small Functions Using Examples

We would like to write code that is correct, and have evidence on hand that it is correct. It can help with both issues is to think about the "size" of a functions. Not in the sense of the amount of code that implements a function, although that is interesting, but rather the size of the mathematical function that our code manifests.

For example, in the game of Go there is a condition called atari in which a player's stones may be captured by their opponent. The rule is that a stone with two or more free spaces adjacent to it (these are called liberties) is not in atari. While it can be surprisingly tricky to count how many liberties a stone has, once we know that working out if the stone is in atari is easy. We might begin by writing a function (in pseudo–code) like this:

```boolean atari (int libertyCount)
if liberty < 2
true
false
```

It might not look it, but this function is pretty huge. A mathematical function can be understood as a set, specifically the Cartesian product of all the sets that are its domain and range. Here the domain is the set of all `int` values and the range is the set of all `boolean` values. If these were the same size as in Java then there would be `2L * (Integer.MAX_VALUE + (-1L * Integer.MIN_VALUE) + 1L)` or 8,589,934,592 members of the set `int * boolean`. The function `atari` as written is trivial, but because the function it models is so big it would be very hard to produce strong evidence that it is correct. It would be ludicrous to set about creating 8.6×109; test cases. This is the essence of the claim that test cannot prove the absence of bugs.

Tests can demonstrate the presence of features, though. But still we have the problem of size.

The problem domain helps us out. The number of liberties is not any integer, or even any `int`, but exactly one of {1,2,3,4}.