Question

The relation R in the set $\left\{ 1,2,3 \right\}$ given by $R=\left\{ \left( 1,2 \right),\left( 2,1 \right) \right\}$ is:
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Reflexive and symmetric

Answer 1

Hint: Think of the basic definition of the types of relations given in the figure and try to check whether the relation mentioned in the questions satisfies the condition for any type of relation or not.

Complete step-by-step solution -
In a given question, we are given the set $\left\{ 1,2,3 \right\}$. On this set a relation is defined as, $R=\left\{ \left( 1,2 \right),\left( 2,1 \right) \right\}$.
Now, this relation will be reflexive if for all the elements of set $\left\{ 1,2,3 \right\}$, that is 1,2 and 3 the ordered pairs of this elements itself, that is $\left( 1,1 \right),\left( 2,2 \right)$ and $\left( 3,3 \right)$ will belong to this relation.
But here, $\left( 1,1 \right),\left( 2,2 \right)$ and $\left( 3,3 \right)$ does not belong to a given relation R. Therefore, R is not reflexive.
Now, given a relation R will be symmetric, if for each ordered pair which belongs to R, the ordered pair with interchange position of elements in the ordered pair also belongs to that relation.
Here ordered pairs in R are $\left( 1,2 \right)$ and $\left( 2,1 \right)$. For $\left( 1,2 \right)$, an ordered pair with an interchanged position of elements is $\left( 2,1 \right)$, which belongs to R. Therefore, R is symmetric.
Now, given relation R will be transitive, if for $\left( x,y \right)$ and $\left( y,z \right)$ belongs to R, where x, y is element of $\left\{ 1,2,3 \right\}$, the ordered pair $\left( x,z \right)$ also belongs to R.
Here, for $\left( 1,2 \right)$ and $\left( 2,1 \right)$ belong to R, $\left( 1,1 \right)$ cannot belong to R.
Therefore, R is not transitive.
Hence, the correct answer is option (b).

Note: In this type of question, where we can write tabular form of relation it is finite, we can directly check conditions from tabular form. Need to remember the definitions of reflexive,symmetric and transitive relation.