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The relation S is defined on the set of integers Z as $z{\text{S}}y$. If integer $z$ divides integer $y$, thenA.S is an equivalence relationB.S is only reflexive and symmetricC.S is only reflexive and transitiveD.S is only symmetric and transitive

Last updated date: 20th Jun 2024
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Hint: First, we will use the definitions of that reflexive means $z$ is the same means both the integers are same, $z{\text{S}}z$, symmetry means if $z{\text{S}}y$ is true then $y{\text{S}}z$ is also true and transitive means that if $z{\text{S}}y$ and $y{\text{S}}x$ exist then $z{\text{S}}x$ also exists to choose the correct option.

We are given that the relation S is defined on the set of integers Z as $z{\text{S}}y$ such that integer $z$ divides integer $y$.
We know that reflexive means $z$ is the same means both the integers are same, $z{\text{S}}z$, symmetry means if $z{\text{S}}y$ is true then $y{\text{S}}z$ is also true and transitive means that if $z{\text{S}}y$ and $y{\text{S}}x$ exist then $z{\text{S}}x$ also exists.
First we will check whether the relation S is symmetric or not.
If we divide an integer by itself then it is always divisible, so we have
$\Rightarrow \dfrac{z}{z} = 1$
Thus, $z{\text{S}}z$ is true, as $z$ always divides $z$.
Hence, S is a reflexive relation.
Now we check whether the relation S is symmetric or not.
If we divide an integer $z$ divides $y$, but $y$ is not related to $z$ as $y$ does not divide $z$.
Hence, S is not a symmetric relation.
Now check whether the relation S is transitive or not.
If $z{\text{S}}y$ is true such that $z$ divides $y$ and $y{\text{S}}x$ is true such that $y$ divides $x$.
So, we have that $z{\text{S}}x$ is true, as $z$ always divides $x$.
Hence, S is a transitive relation.
So, we have found out that S is only reflexive and transitive but not symmetric. So, it is not an equivalence relation.
Hence, option C is correct.

Note: We know that an equivalence relation means if we have reflexive relation, symmetric relation and transitive relation. Only when all the three relations are satisfied then the relation is an equivalence relation. For example, we also need to know that the 4 divides 2 but 2 does not divide 4 for the symmetric relation.