Expert: Sherry Wallin - 3/22/2009

Question
I have a question on equivalence relations that I do not know how to start answering....can you help please?

Let relation R be reflexive and transitive on a set S.

Let aPb iff aRb and bRa.  Show P is an equivalence relation on S.

Answer
Steve~
   To be an equivalence relation on a set S you  must show that the relation is reflexive, symmetric and transitive. Your goal then is to use your givens to show: if aPb then bPa (symmetric), aPa (reflexive), and if aPb and if bPc then aPc (transitive). You already are given that R is reflexive and transitive on S. Further you are told that aPb if aRb and bRa, so show through the reflexive property and transitive property on R that you also have the symmetric property which gives you that aPb.

From Let aPb iff aRb and bRa we get:
Suppose aPb, then aRb and bRa so bRa and aRb,therefore bPa, thus P is symmetric. Now aPb and bPa gives us aPa, thus P is reflexive. [also bRb by the same reasoning]. If aPb and bRb and bRa then P is transitive. Therefore P is an equivalence relation.