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.