Question

Let X be the set of all citizens of India. Elements x, y in X are said to be related if the difference of their age is 5 years. Which one of the following is correct?
A) The relation is an equivalence relation on X.
B) The relation is symmetric but neither reflexive nor transitive
C) The relation is reflexive but neither symmetric nor transitive
D) None of these

Answer 1

Hint- In order to solve the problem use the property of reflexive, symmetric, transitive and equivalence relation. Separately check for all of them before finalizing the answer.

Complete step-by-step answer:
Given that:
X = {All the citizens of India}
Here x is a set representing all the citizens of India.
And $R = \left\{ {\left( {x,y} \right):x,y \in X,\left| {x - y} \right| = 5} \right\}$
We have been given a relation R as described above where the difference of the age is 5 years.
Now we have to find out the type of relationship of “R”
Let us first check for reflexive nature.
As we know that for Reflexive relation:
$\left| {a - b} \right| = 0$ is the criterion but we have $\left| {x - y} \right| = 5$
So, the relation R is not reflexive.
Now let us check if the relation is symmetric or not.
As we know that for symmetric relation:
The criterion is $\left| {a - b} \right| = \left| {b - a} \right|$
As for the given case we have:
$\left| {x - y} \right| = \left| {y - x} \right| = 5$
So, R is a symmetric relation.
Now let us check if the relation is transitive or not.
For transitive relation we have:
If $\left| {a - b} \right| = t\& \left| {b - c} \right| = t$ we must have $\left| {c - a} \right| = t$
For the given case let $\left| {x - y} \right| = 5\& \left| {y - z} \right| = 5$
As for our case we have
$
  \left| {x - y} \right| = 5\& \left| {y - z} \right| = 5 \\
   \Rightarrow \left| {z - x} \right| = - \left( {\left| {x - y} \right| - \left| {y - z} \right|} \right) \\
   \Rightarrow \left| {z - x} \right| = - \left( {5 - 5} \right) \\
   \Rightarrow \left| {z - x} \right| = 0 \\
 $
As we have$\left| {z - x} \right| = 0$ and not equal to 5
So the given relation is not transitive.
As the relation is not reflexive and transitive so the relation will not be equivalence.
Hence, the relation is symmetric but neither reflexive nor transitive.
So, option B is the correct option.

Note- There are 9 types of relations in maths namely: empty relation, full relation, reflexive relation, irreflexive relation, symmetric relation, anti-symmetric relation, transitive relation, equivalence relation, and asymmetric relation. An equivalence relation is a binary relation that is reflexive, symmetric and transitive.