Question

Define a relation $R$ on the set $N$ such that $R = \left\{ {\left( {x,y} \right):2x - y = 10} \right\}$ . Find the correct option from below about $R$ .
1) $R$ is Reflexive
2) $R$ is Symmetric
3) $R$ is Transitive
4) None of the above

Answer 1

Hint: For this question, we have to go by option wise. At first, we will check whether it is Reflexive or not and then we go for Symmetry and then we go for Transitive. We will determine $R$ by the definitions of reflexive, symmetry, and transitive.

Complete step-by-step answer:
Reflexive: If $\left( {x,x} \right)$is in relation to $R$ for every $x$ then $R$ is Reflexive.
Let us check the definition for given $R = \left\{ {\left( {x,y} \right):2x - y = 10} \right\}$.
Let us substitute $\left( {x,x} \right)$ in place of $\left( {x,y} \right)$ to determine whether $R$ is Reflexive or not.
$2x - y = 10 \Rightarrow 2x - x = 10$
$ \Rightarrow x = 10$.
So, it is only true for $x = 10$ .
Therefore $R$is not Reflexive since it is not true for all $x$.
Symmetry: If $$\left( {x,y} \right) \in R \Rightarrow \left( {y,x} \right) \in R$$ then $R$ is Symmetric.
Let us assume $$\left( {x,y} \right) \in R \Rightarrow 2x - y = 10$$.
Now we have to check whether $$\left( {y,x} \right) \in R$$or $$\left( {y,x} \right) \notin R$$
But if, $$\left( {y,x} \right) \in R \Rightarrow 2y - x = 10$$,
Since $2x - y = 10$ does not imply $2y - x = 10$.
Therefore $R$ is not Symmetric.
Transitive: If $\left( {x,y} \right),\left( {y,z} \right) \in R \Rightarrow \left( {x,z} \right) \in R$ then $R$ is Transitive.
Let us assume $\left( {x,y} \right),\left( {y,z} \right) \in R \Rightarrow 2x - y = 10,2y - z = 10$.
Now we have to check whether $$\left( {x,z} \right) \in R$$or $$\left( {x,z} \right) \notin R$$
But if, $$\left( {x,z} \right) \in R \Rightarrow 2x - z = 10$$,
Since $2x - y = 10,2y - z = 10$ does not imply $2x - z = 10$.
Therefore $R$ is not Transitive.
So, the final answer is option 4 None of the above.
So, the correct answer is “Option 4”.

Note: We have to be careful while determining whether it is reflexive or not because it may satisfy some $x$ but the definition says that it should satisfy all the values of $x$ . There is another relation called equivalent: it is nothing but satisfying all the three properties namely reflexive, symmetry, and transitive. If these three properties satisfy the relation then that relation is called an equivalence relation.