Question

For $n,m\in N\ n/m$ means that n is a factor of m, the relation / is,
A. Reflexive and symmetric
B. Transitive and symmetric
C. Reflexive transitive and symmetric
D. Reflexive transitive and not symmetric

Answer 1

Hint: We will be using the concepts of functions and relations to solve the problem. We will be using the definitions of reflexive relation, symmetric relations and transitive relations to verify if each relation holds or not and hence deduce the answer.

Complete step-by-step answer:

Now, we have been given a relation and we have to find whether the relation is reflexive, symmetric, transitive or a combination of these.
Now, we know that reflexive relations are those in which every element is mapped to itself i.e. $\left( a,a \right)\in R$ while symmetric relations are those for which if a R b then b R a. Also, holds and transitive are those relations in which if a R b and b R c then a R c must be held.
Now, we know different types of relations, we will check the given relation for these.
Now, we have been given a relation R, such that $n,m\in N\ n/m$ means that n is a factor of m.
Now, for Reflexive we have for $a\in N\ a\left| a \right.$ hold as a is a factor a that is every number is a factor of itself.
Now, for symmetric we have $a,b\in N$such that if,
a/b that is a is factor of b.
Now, b/a that is b is a factor of a will not hold in general like if we take (a, b) as (2, 6) then 2 is a factor of 6 but 6 is not a factor of 2. Therefore, $aRb{\nRightarrow }bRa$. Hence, not symmetric.
Now, for transitive we have a,b,c in N . Now, if a R b that is a/b or a is a factor of b and b R c that is b/c or b is a factor of c then a R c that is a/c or a is a factor c holds always. For example, if we have a, b, c as 2, 4, 8 then 2 is a factor of 4 and 4 is a factor of 8 then 2 is also a factor of 8.
Hence, the relation a/b is transitive also.
Now, the correct option is (D). Since, the relation is reflexive, transitive but not symmetric.

Note: To solve these types of questions it is important to note that a R b means that a is related to b by a relation R. Also these types of questions are solved easily by giving examples and counterexamples.