## Mathematics behind Comparison #2: Ordering Relations in Math

In order to sort a collection of elements you need to provide a sorting predicate that determines when one element is less than the other. This predicate must “induce a strict total ordering on the equivalence classes” according to cppreference. Wait, what?

The upcoming C++ spaceship operator implements a three way comparison,
i.e. it is a single function that can return the results of `<`

, `==`

and `>`

combined.
But related to it are terms like “strong equality” and “weak ordering” which are somewhat confusing if you don’t have the mathematical background.

So let’s untangle it: This series will explain both the mathematics behind equality and ordering, as well as give concrete guidelines for implementing the comparison operators and the spaceship operator.

This part covers the mathematics behind ordering relations. They are a lot more complicated than equivalence relations we’ve looked at before. As my blog posts are usually long anyway, I’ve decided to split it into two. So this part is only about the mathematics while the next part—already published—is about how they should be implemented in C++.

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