The relation S is defined on the set of integers Z as xSy if integer x divides y. We will check one by one for each property. Reflexivity: For any integer a ∈ Z the integer always divides itself with a. Therefore, we have aSa. That means the relation S is reflexive. Symmetry: We will prove that the given relation is not symmetric. Consider the integers 4 and 8. We know that 4 divides 8 but 8 does not divide 4. Therefore, the given relation is not symmetric. Transitivity: Consider three integers a, b and c. Assume that aRb and bRc that means a divides b and b divides c. We will prove that a divides c We know that a divides b that means for a constant k1 we can write b=k1a. Similarly, if b divides c that means for a constant k2 we can write c=k2b. We have to find a constant k such that c=ka We already know that c=k2b and b=k1a We can write c=k2(k1a). As k1 and k2 are constants put k=k1k2. Therefore, c = ka. Therefore, a divides c, implies aRc. Therefore, the given relation is transitive. From all the three points we can say that the relation is not an equivalence relation.