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.
Motivations: 1