# How can I write these things formally?

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 ? :/

I guess, you have a set $$C$$ of $$|C|=N$$ computers endowed with a total order $$\le$$.
• How can I write what $C$ is ? The way I did it in my question, is it correct? If yes I can add your answer to my notation because your answer is dealing with rankable what I was missing and that each computer is different. I mean, I'm having trouble to see how it's supposed to look like overall. Dec 6, 2020 at 12:54
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.