4.2 Equivalence relations
Our formal definition of an equivalence relation involves three key properties. A relation that has these three properties partitions the set on which the relation is defined, as we show later in this subsection.
The reflexive, symmetric and transitive properties are independent, in the sense that relations exist with every combination of these properties. (However, relations which are symmetric and transitive but not reflexive are usually somewhat contrived.)
If a relation is symmetric, then ‘x is related to y’ means the same as ‘y is related to x’, and we can use either phrase, or simply say ‘x and y are related’; we can write either x y or y x.
We now consider the examples in the previous subsection to see whether they satisfy any or all of the three properties.