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Relations

Relations describe the relationships between different elements of the same set or different sets. They are another fundamental concept in mathematics, for instance, they provide a foundation of functions, which turn out to be special cases of relations.

We will see in this part of BookofProofs that, from the set-theoretical point of view, relations are sets. As such, being the set element $”\in”$ or being the subset $”\subseteq”$ are not relations, but predicates. Loosely speaking, they are relations in a metalanguage, which deals with sets.

Apart from these “metarelations”, there are many examples of “usual” relations we deal with in mathematics, for instance order relations $”\ge”$, the perpendicularity $”\perp”$ of straight lines in a plane, the equivalence relations and many, many more.