# Mathematics behind Comparison #5: Ordering Algorithms

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.

To finish this series let’s talk about algorithms that require an ordering and how they can be implemented using three-way comparison.

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