Question

Let R be a relation on the set N of natural numbers defined by nRm ⇔ n is a factor of m(i.e., n|m) . Then R is
1. ${\text{ }}Reflexive{\text{ }}and{\text{ }}symmetric$
2. ${\text{ }}Transitive{\text{ }}and{\text{ }}symmetric$
3. ${\text{ }}Equivalence$
4. ${\text{ }}Reflexive,{\text{ }}transitive{\text{ }}but{\text{ }}not{\text{ }}symmetric$

Answer 1

We have to find the relation R . We solve this using the concept of relation and function . We should have knowledge of the relation or of the function . We should have the knowledge of types of relations and we should know how to prove the types of relations . We should also know about equivalence relations .

Complete step-by-step solution:
Given :
The set N of natural numbers defined by $n{R}m \Leftrightarrow n$ is a factor of m(i.e., $n|m$ )
Types of relations :
(1) reflexive relation
(2) symmetric relation
(3) transitive relation
To check the relation between the relations :-
For reflexive relation :
If $\left( {n{\text{ }},{\text{ }}n} \right) \in R$ , for every $n \in N$
$n|n{\text{ }}:{\text{ }}n$ is a factor of $n$
Hence the relation is reflexive
For symmetric relation :
If $\left( {{\text{ }}n{\text{ }},{\text{ }}m{\text{ }}} \right) \in R$ implies that $\left( {{\text{ }}m{\text{ }},{\text{ }}n{\text{ }}} \right) \in R$ for all $n{\text{ }},{\text{ }}m \in N$
$n|m{\text{ }}:{\text{ }}n$ is a factor of $m$
So , it does not implies that
$m|n{\text{ }}:{\text{ }}m$ is a factor of $n$
As , both numbers can’t be factors of each other .
Hence the relation is not symmetric .
For transitive relation :
If $\left( {{\text{ }}n{\text{ }},{\text{ }}m} \right) \in R$ and $\left( {m{\text{ }},{\text{ }}o} \right) \in R$ implies that $\left( {{\text{ }}n{\text{ }},{\text{ }}o} \right) \in R$ for all $n{\text{ }},{\text{ }}m{\text{ }},{\text{ }}o \in N$
$n|m{\text{ }}:{\text{ }}n$ is a factor of $m$
$m|o{\text{ }}:{\text{ }}m$ is a factor of $o$
So , it implies that
$n|o{\text{ }}:{\text{ }}n$ is a factor of $o$
Thus , $n{\text{ }},{\text{ }}o \in R$ for all values of $n{\text{ }},{\text{ }}o \in N$
Hence the relation is transitive .
Thus , the relation $R$ is reflexive and transitive but not symmetric .
Hence , the correct option is $\;\left( 4 \right)$ .

Note: A relation R in a set A is called empty relation , if no element of A is related to any element of A I.e. $\;R{\text{ }} = {\text{ }}\emptyset \subset A{\text{ }} \times {\text{ }}A{\text{ }}.$
A relation R in a set A is called a universal relation , if each element of A is related to any element of A I.e. $R{\text{ }} = {\text{ }}A{\text{ }} \times {\text{ }}A$ .
Both the empty relation and the universal relation are sometimes called trivial relations .