Question

The relation S is defined on the set of integers Z as zSy if integer z divides integer y. Then.
A) S in an equivalence relation
B) S is only reflexive and symmetric
C) S is only reflexive and transitive
D) S is only symmetric and transitive

Answer 1

Hint:
In this question, they have given us the relation between z and y is denoted by S and is defined as zSy. It means that z integer divided integer y.
We have to choose the correct option. If the relation S is reflexive, transitive as well as symmetric then relation S is called an equivalence relation. Check whether the relation S is reflexive or not, it is transitive or not and it is symmetric or not.

Complete step by step solution:
We have given the relation S which is defined on the set of integers Z as zSy which defines that integer z divides integer y.
First of all, check whether the relation S is reflexive or not. Reflexive means z is the same means both the integers are same i.e. Z=y. if we divide an integer by itself then it is always divisible.
$ \Rightarrow \dfrac{z}{z} = 1$
$ \Rightarrow zSz$is true, as z always divides z.
So, S is reflexive relation
Now check whether S is symmetric or not. By Symmetry, it means if zSy is true then ySz is also true. Symmetry means it is defined when we interchange the integers.
$ \Rightarrow y\,\text{is not related to}\,z$, as y doesn’t divide z.
Here, if z divides y then it is not necessary that y divides z e.g. 6 divides 48 but 48 doesn’t divide 6.
So, S is not symmetric.
Check for transitive: Transitive means that if we have three integers and the relation of them should exist i.e. if zSy and ySx exist then zSx also exists. Here, if z divides y and y divides x then z divides x also e.g. 2 divided 6 and 6 divides 48 and hence 2 divides 48.
$ \Rightarrow zSy$
 $ \Rightarrow ySx$
$ \Rightarrow zSx$is true, as z also divides x.
So, S is a transitive relation.
In the end, it is concluded that S is only reflexive and transitive but not symmetric. So, it is not an equivalence relation.

Hence, option C is the correct answer.

Note:
An equivalence relation means if we have two elements and we have to identify whether they share a set of common attributes. So, when we talk about equality relation, equality satisfies the reflexive i.e. z=z for all values of z, transitive i.e. z=y implies y=z and symmetric i.e. z=y and y=x implies z=x properties.