Relations

Abstract

This chapter introduces relations between sets with an emphasis on binary relations. Formally, n-ary relations are defined as sets of n-tuples, hence binary relations as sets of ordered pairs. In the latter case, the notion of source and target sets is introduced and a distinction between general heterogeneous relations and homogeneous relations, where the source and target set are the same, is made. Graphical ways of presenting relations are introduced and it is shown that functions arise as special cases of binary relations through their graphs. Next, various operations on binary relations are explained; beyond Boolean operations such as union, intersection and complementation these are the composition of relations, the converse of a relation and the domain and range of a relation. After that, special binary relations, such as reflexive and irreflexive relations, symmetric and antisymmetric relations as well as transitive relations are introduced. They are then used to define order and equivalence relations, as well as advanced notions such as equivalence classes and partitions.