Question

Let $R = \left\{ {\left( {1,3} \right),\left( {4,2} \right),\left( {2,4} \right),\left( {2,3} \right),\left( {3,1} \right)} \right\}$ be a relation on the set $A = \left\{ {1,2,3,4} \right\}$. The relation R is:
A. A function
B. Transitive
C. Not symmetric
D. Reflexive

Answer 1

Hint- We will check each option by keeping in mind the conditions making them eligible to be called transitive etc. After doing so, we will get the correct answer(s) out of four which will be present in the above options.

Complete step-by-step answer:
In the question, it is given that-
$ \to A = \left\{ {1,2,3,4} \right\}$
The relation given to us is-
$ \to R = \left\{ {\left( {1,3} \right),\left( {4,2} \right),\left( {2,4} \right),\left( {2,3} \right),\left( {3,1} \right)} \right\}$
First, we will check if the given relation is reflexive or not. If the relation given is reflexive, it should have $\left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {4,4} \right)} \right\}$.
When we check in the relation R if it has $\left\{ {\left( {1,1} \right),\left( {2,2} \right),\left( {3,3} \right),\left( {4,4} \right)} \right\}$, we come to notice that it doesn’t. which proves that the given relation R is not reflexive.
Thus, option D is not the correct option.
Second, we will check if the relation given to us is symmetric or not. If it is symmetric, it should have $\left\{ {\left( {1,3} \right),\left( {3,1} \right),\left( {4,2} \right),\left( {2,4} \right),\left( {2,3} \right),\left( {3,2} \right)} \right\}$.
 For the above relation to be symmetric, only $\left\{ {\left( {3,2} \right)} \right\}$is missing. Thus, the above option is not symmetric.
Now, Not Symmetric is given in the options. But we must check if other options are correct or not because there is always a possibility for more than one option to be correct.
Third, we will check for the above relation to be transitive. If it is transitive, $\left\{ {\left( {4,2} \right),\left( {2,4} \right),\left( {4,4} \right)} \right\}$. Since it doesn’t have $\left\{ {\left( {4,4} \right)} \right\}$, it is not transitive.
Fourth, we will check if it is a function. Since it is given in relation, it is not a function.
So, only option C is correct.

Note: Do not leave the answer without solving all the parts in such questions. If there are more than one correct answer, your answer might be marked as wrong. Remember the properties/conditions while solving the answer.