Question

If $n\left| m \right.$ means n is a factor of m, the relation $\left| {} \right.$ is:
A. Reflexive and symmetric
B. Transitive and symmetric
C. Reflexive, transitive and symmetric
D. Reflexive, transitive, and not symmetric

Answer 1

In this question, we need to find the type of relation of ‘$\left| {} \right.$’ which denotes that $n\left| m \right.$ means n is a factor of m. To check if the relation is reflexive, we need to check if $n\left| n \right.$ i.e. if n is a factor of n. To check if the relation is symmetric, we need to check that $n\left| m \right.$ implies $m\left| n \right.$. To check if the relation is transitive, we need to check if $n\left| m \right.$ and $m\left| p \right.$ implies $n\left| p \right.$.

Complete step-by-step solution
Here, we are given the relation $\left| {} \right.$ according to which $n\left| m \right.$ means n is a factor of m. We need to find the type of relation $\left| {} \right.$.
For reflexive: We need to show that $n\left| m \right.$. Hence, we need to show that n is a factor of n. As we know, every number divides itself, therefore, every number is a factor of itself. Hence, $\left| {} \right.$ is a reflexive relation.
For symmetric: We need to show that if $n\left| m \right.$ then this implies $m\left| n \right.$. Hence, we need to show that, if n is a factor of m, then m is a factor of n.
As we know that, if n is a factor of m, then m is not always a factor of n. For example, if 2 is a factor of 4, then 4 is not a factor of 2.
Hence, relation $\left| {} \right.$ is not a symmetric relation.
For transitive: We need to show that, if $n\left| m \right.$ and $m\left| p \right.$ then $n\left| p \right.$. Hence, we need to show that, if n is a factor of m and m is a factor of p then n is a factor of p.
Since n is a factor of m, therefore $n\cdot k=m$ for some k as an integer. Also, since m is a factor of p, therefore $m\cdot l=p$ for some l as an integer.
From $ml=p,m=\dfrac{p}{l}$. Putting the value of m into $nk=m$ we get $nk=\dfrac{p}{l}\Rightarrow nkl=p$.
Since k and l are integers, so kl is also an integer.
Hence, n is a factor of p. Hence $n\left| p \right.$.
Therefore, relation $\left| {} \right.$ is transitive.
Hence, the relation is reflexive, transitive, and not symmetric.
So, option D is the correct answer.

Note: Students can get confused between reflexive and symmetric relations. Note that, if a relation is reflexive, symmetric, and transitive, then the relation is said to be an equivalence relation. In mathematical form, we define relation as follows:
(I) Reflexive: $\forall a\in A$ if aRa then relation R is a reflexive relation.
(II) Symmetric: $\forall a,b\in A$ if $aRb\Rightarrow bRa$ then relation R is a symmetric relation.
(III) Transitive: $\forall a,b,c\in A$ if aRb and bRc $\Rightarrow aRc$ then relation R is transitive relation.