# Mathematics behind Comparison #3: Ordering Relations in C++

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.

The previous part was very math heavy but necessary: It introduced the mathematical terminology for ordering relations. With that done we can finally talk about how it applies to C++.

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