I am trying to understand the following math expression, describing the possible sets of a 15 Puzzle:
The expressions for tile and position are straight forward, I am having a difficulty "translating" the board expression to plain English.
Here is what I understand:
Board is a set of a combination of all positions C and error. C is defined as the set of all combinations positions and with all tiles. C is always 16 (restriction given by |).
And (^) for all c (a combination), which is defined by p (position) and t which are subset of C we must also have: All c', defined by p' and t' there is the requirement that c' is different then c and hence p' and p are not equal and also t' and t are not equal. Meaning each tile can be on a single location of each combination.
This is horribly complicated in English, compared to math. However, I am not sure I understand it correctly, and I would be happy to hear if I am wrong or right, and how the above expression can be interpreted.
Ok, let's get through this step by step: The set $(\mathrm{position}\setminus\mathrm{error})\times\mathrm{tile}$ is the set of all tuples $(p,t)$ for which $p$ is a position, and $t$ is a tile. Such a tuple just says that the tile $t$ is currently at position $p$.
A set $C$ is just a subset of $(\mathrm{position}\setminus\mathrm{error})\times\mathrm{tile}$ with certain restrictions. First of all, it has to have exactly 16 elements. Secondly, if we take to elements $c$, $c'$ of $C$, which are different to each other we want them to differ in both coordinates. So if we have two different positions, we want to associate two different tiles. And vice versa, given two different tiles, they have to be at two different positions.
The set $\mathrm{board}$ is now the union of the set of all $C$ that satisfy theses conditions, i.e. the set of all valid boards, with the single-element set $\{\mathrm{error}\}$. So $\mathrm{board}$ contains all valid board positions and a single board position $\mathrm{error}$ to denote non-valid boards.