Relations on Sets

This paper is dedicated to the memory of George Boole FRS (1815-1864), first Professor of Mathematics at University College Cork. A relation on a non-empty set K is any subset W of the cartesian product K × K. If (x, x) ∈ W for every x ∈ K, W is said to be reflexive on K. If for x, y, z ∈ K, (x, y) ∈ W implies that (y, x) ∈ W, W is said to be symmetric on K. If for x, y, z ∈ K (x, y) ∈ W and (y, z) ∈ W together imply that (x, z) ∈ W, W is said to be transitive on K. Finally, if the relation W is reflexive, symmetric and transitive on K, W is said to be an equivalence relation on K. We contend that equivalence relations are of central importance in many areas of mathematics. Many texts on algebra (for example Topics in Algebra [1]) contain as an exercise the following fallacious argument which the reader is challenged to refute.