Question

For $n,m\in N,n|m$ means that n is a factor of m, then relation | is
(A) Reflexive and symmetric
(B) Transitive and symmetric
(C) Reflexive, transitive and symmetric
(D) Reflexive, transitive and not symmetric

Answer 1

Hint: First of all, check the relation | for transitive relation. Take another number $l$ such that $l\in N$ such that $m$ is a factor of $l$ . It is given that $n,m\in N,n|m$ means that $n$ is a factor of $m$ . Now, check whether $n$ is a factor of $l$ or not. Then, decide whether the relation | is transitive or not. For the relation | to be symmetric, $m$ must also be a factor of $n$ . As $n$ and $m$ belong to the natural number and $n|m$ means that n is a factor of m but it is not necessary that $m$ is also a factor of $n$ .

Complete step-by-step answer:
Now, check for reflexive. For reflexivity, n must be a factor of itself.
According to the question, it is given that $n,m\in N,n|m$ means that $n$ is a factor of $m$ . We have to find the nature of the relation |.
Let us take another number $l$ such that $l\in N$ such that $m$ is a factor of $l$ .
The relation $m|l$ means that m is a factor of l ……………………………….(1)
The relation $n|m$ means that n is a factor of m …………………………………..(2)
Now, from equation (1) and equation (2), we can say that $n$ is also a factor of $l$ .
As $n$ is a factor of $l$ so, $n|l$ ………………….(3)
From equation (1), equation (2), and equation (3), we can say that the relation | is transitive ……………………………(4)
As n and m belong to the natural number and $n|m$ means that n is a factor of m. For the relation | to be symmetric, $m$ must also be a factor of $n$ . Since $n$ is a factor of m but it is not necessary that $m$ must also be a factor of $n$ . It means that the given relation is not symmetric.
Therefore, the relation | is not symmetric ……………………………..(5)
Since $n$ is also a factor of $n$ so, we can say that $n$ is reflexive, $n|n$ ………………………………..(6)
From equation (4), equation (5), and equation (6), we can say that the relation | is reflexive, transitive, and not symmetric.
Hence, option (D) is the correct one.

Note: We can also solve this question without checking it for reflexive relation and transitive relation. Here, we can pick the correct option only by checking the relation |, whether it is symmetric or not. Since the relation | is not symmetric and only option (D) states that the relation | non-symmetric. So, option (D) is the correct one.