How can I write these things formally?

Question

Let's say we have $N$ computers of different kind given (every computer is different and only exists once). They are rankable, so you can order them from best to worst computer. There is only $1$ best computer.

How can this be written formally, as a set ?

I've thought about expressing these $N$ computers as an indexed set, so $\left\{1,2,..,N\right\}$

Then , we need our computer set: $$C=\left\{c_i \mid i \in \left\{1,..,N\right\}\right\}$$

But now I have problem writing the best computer, can it be done like this? : Let $c_i$ with $i=1$ be the best computer.

And also I'm having problem with my set above because I'm not sure if it's really rankable the way I have written it, and also I don't know how to write it such that each of its elements is distinct / different to all its other elements ? :/

Answer 1

I guess, you have a set $C$ of $|C|=N$ computers endowed with a total order $\le$.

Answer 2

The set of distinct computers could be denoted by $C \ =\ \{C_1,C_2,...,C_n\}$. The set of ranks has to have positive integers only. So instead of $R$, we could use positive integers $Z_{+}$. Since every computer is unique, it has a different rank than the other and let's suppose it is possible to rank them using some method. I am thinking of a one-to-one function (injection) from the set of computers to the set of positive integers. Then this system of computers and their ranking property could be written as $f:C\rightarrow Z_{+}$ where $f$ is a function to rank them and $C$ and $Z_{+}$ are the set of computers and positive integers respectively.