Question

Let R be a relation on the set N of natural numbers defined by $nRm$. If $n$ divides $m$. Then, $R$ is
A. Reflexive and symmetric
B. Transitive and symmetric
C. Equivalence
D. Reflexive, transitive but not symmetric

Answer 1

Hint: In order to solve this question, we have to use the properties to determine whether a relation is reflexive, symmetric, transitive or equivalence. Thus, we will get our desired answer.

Complete step-by-step answer:

For Symmetric relation,
 Symmetric Relation - A relation is said to be symmetric if a, b belongs to R b, a also belongs to R.
$
  {\text{ }}R = \left\{ {\left( {a,b} \right){\text{ | a}} \div {\text{b}}} \right\} \\
   \Rightarrow \left( {2,4} \right) \leftarrow R \Rightarrow {\text{ 2}} \div {\text{4}} \\
 $
But 4 does not divided by 2 that is (4,2) does not belongs to R
$\therefore $ R is not symmetric
For Reflexive,
Reflexive relation – A relation R over a set X is reflexive if it relates every element of X to itself
For all $n$ belongs to $N$,$n$divided by $n$
$
  \forall {\text{n}} \leftarrow {\text{N,n}} \div {\text{n}} \\
   \Rightarrow \left( {n,n} \right) \leftarrow R{\text{ }}\forall {\text{n}} \leftarrow {\text{N}} \\
 $
Hence, R is a reflexive relation.
For transitive relation,
Transitive relation – If for all elements a, b, c in X whenever, R relates a to b and b to c, then R also relates a to c. Transitive is a key property of both partial orders and equivalence relations.
Let,
 (If $a \div b{\text{ and b}} \div {\text{c then, a}} \div {\text{c}}$)
If $\left( {a,b} \right) \leftarrow R,\left( {b,c} \right) \leftarrow R$ (If $\left( {a,b} \right)$ belongs to $R$, then $\left( {b,c} \right)$ also belongs to $R$)
For,$\left( {a,b} \right)$$ = a \div b$,
For, $\left( {b,c} \right)$=$b \div c$
$
   \Rightarrow a \div c \\
   \Rightarrow \left( {a,c} \right) \leftarrow R \\
 $
Therefore, $R$ is a transitive relation.

For equivalence relation,
A relation between elements of a set which is reflexive, symmetric, and transitive and which defines exclusive classes whose members bear the relation to each other and not to those in the other classes.
Therefore $ R $ is not an equivalence relation.
Hence, option D is correct.

Note: Whenever we face such types of questions, the key concept is that we have to determine all the relations by their properties. Like in this question, by using properties we come up with a solution that the relation is reflexive, transitive but not symmetric. Thus, we get our required answer.