Question

Let X be the set of all persons living in Delhi. The persons a and b in X are said to be related if the difference in their ages is at most 5 years. The relation is
A. An equivalence relation
B. Reflexive and transitive but not symmetric
C. Symmetric and transitive but not reflexive
D. Reflexive and symmetric but not transitive

Answer 1

Hint: In the given question, we are given a set X having all people living in Delhi. To find the relation, we will check the equivalence of the set with the given condition, that is, we will check for reflexive, symmetric and transitive in the given set. We will first define the set X and the relation R as per the given condition, that is, \[R=\{(a,b):\left| a-b \right|\le 5\}\] . For reflexive, we will use the element \[(a,a)\]. To check for symmetry, we have to prove if \[\left( a,b \right)\in R\] then \[\left( b,a \right)\in R\]. And for transitive, we will check if \[\left( a,b \right)\in R\] and \[\left( b,c \right)\in R\] then \[\left( a,c \right)\in R\]. Hence, we will have the relation for the given condition.

Complete step by step solution:
According to the given question, we are given a set named X which is the set of all people living in Delhi. The persons are related only if their ages differ by at most 5 years and we have to find the correct option from the given.
So, we have,
\[X=\{People\text{ }living\text{ }in\text{ }Delhi\}\]
Let the relation between the people in Delhi be represented by R, we have,
\[R=\{(a,b):\left| a-b \right|\le 5\}\]
So, we will check the equivalence relation and for that we will check for reflexive, symmetric and transitive in the given set.
For reflexive, let \[(a,a)\in R\]
We get,
\[\Rightarrow \left| a-a \right|\le 5\]
\[\Rightarrow 0\le 5\]
Since, the condition is correct, the relation R is reflexive.
For symmetric, \[\left( a,b \right)\in R\]
\[\Rightarrow \left| a-b \right|\le 5\]
We can also write it as,
\[\Rightarrow \left| b-a \right|\le 5\]
Which we can write it as,
\[\Rightarrow \left( b,a \right)\in R\]
Since, \[\left( a,b \right)=\left( b,a \right)\], therefore the relation R is symmetric.
For transitive, we have two element \[\left( a,b \right),\left( b,c \right)\in R\] such that,
\[\left| a-b \right|\le 5\] and \[\left| b-c \right|\le 5\]
Adding up the above expression, we get,
\[\left| a-b \right|+\left| b-c \right|\le 10\]
\[\Rightarrow \left| a-c \right|\le 10\]
Which we can write it as,
\[\Rightarrow \left( a,c \right)\le 10\]
And it is not similar to \[\left( a,c \right)\le 5\]
So, \[\left( a,c \right)\notin R\].
Therefore, the given relation R is not transitive.
Therefore, option D. Reflexive and symmetric but not transitive is correct.

Note: The reflexive, symmetric and transitive relation should be carefully checked as they determine if a relation is an equivalence relation or not. The options should be read carefully and should not be confused so that the correct answer is marked.