UPSC NDA Math Model Paper 6

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 $k_{1}$ we can write $b=k_{1} a$. Similarly, if b divides $c$ that means for a constant $k_{2}$ we can write $c=k_{2} b$. We have to find a constant $k$ such that $c=k a$ We already know that $c=k_{2} b$ and $b=k_{1} a$ We can write $c=k_{2}(k_{1} a)$. As $k_{1}$ and $k_{2}$ are constants put $k=k_{1} k_{2}$. 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.