2.6. EQUIVALENCE RELATIONS AND SET PARTITIONS129Now for allx2X,xPxbecause there is somei2Isuch thatx2Xi. The definition ofxPyis clearly symmetric inxandy. Finally,ifxPyandyPz, then there existi2Isuch that bothxandyareelements ofXi, and furthermore there existsj2Isuch that bothyandzare elements ofXj. Nowiis necessarily equal tojbecauseyis anelement of bothXiand ofXjandXi\XjD ;ifi¤j. But then bothxandzare elements ofXi, which givesxPz.Every partition of a setXgives rise to an equivalence relation onX.Suppose that we start with an equivalence relation on a setX, formthe partition ofXinto equivalence classes, and then build the equivalencerelation related to this partition. Then we just get back the equivalencerelation we started with! In fact, letbe an equivalence relation onX, letPD fOExŁWx2Xgbe the corresponding partition ofXinto equivalenceclasses, and letP
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