Analyzing the Relation Based on Age Difference in Delhi

The problem defines a set $X$ which consists of all persons living in Delhi. A relation $R$ is defined on this set $X$. Two persons $a$ and $b$ in $X$ are related if the difference in their ages is at most 5 years. We need to determine the properties of this relation, specifically if it is reflexive, symmetric, and transitive.

Let's analyze each property:

Understanding Reflexivity

A relation $R$ on a set $X$ is said to be reflexive if for every element $a \in X$, $(a, a) \in R$. In the context of this problem, this means that for any person $a$ in Delhi, the difference in age between $a$ and $a$ must be at most 5 years.

The age difference between a person and themselves is always 0. Since $0 \le 5$, the condition holds for every person $a \in X$.

Therefore, the relation is reflexive.

Understanding Symmetry

A relation $R$ on a set $X$ is said to be symmetric if for every pair of elements $a, b \in X$, whenever $(a, b) \in R$, it implies that $(b, a) \in R$. In this problem, if the difference in age between person $a$ and person $b$ is at most 5 years, then the difference in age between person $b$ and person $a$ must also be at most 5 years.

Let the age of person $a$ be $A_a$ and the age of person $b$ be $A_b$. The condition for $(a, b) \in R$ is $|A_a - A_b| \le 5$.

The difference in age between $b$ and $a$ is $|A_b - A_a|$. We know that $|A_b - A_a| = |-(A_a - A_b)| = |A_a - A_b|$.

Since $|A_a - A_b| \le 5$, it directly follows that $|A_b - A_a| \le 5$. So, if $(a, b) \in R$, then $(b, a) \in R$.

Therefore, the relation is symmetric.

Understanding Transitivity

A relation $R$ on a set $X$ is said to be transitive if for every three elements $a, b, c \in X$, whenever $(a, b) \in R$ and $(b, c) \in R$, it implies that $(a, c) \in R$. In this problem, if the difference in age between $a$ and $b$ is at most 5 years, and the difference in age between $b$ and $c$ is at most 5 years, does it necessarily mean the difference in age between $a$ and $c$ is at most 5 years?

Let's consider a counterexample. Suppose we have three persons $a$, $b$, and $c$ with the following ages:

• Person $a$ is 20 years old.
• Person $b$ is 24 years old.
• Person $c$ is 28 years old.

Let's check the relation between these persons:

• Difference in age between $a$ and $b$: $|20 - 24| = |-4| = 4$. Since $4 \le 5$, $(a, b) \in R$.
• Difference in age between $b$ and $c$: $|24 - 28| = |-4| = 4$. Since $4 \le 5$, $(b, c) \in R$.

Now, let's check the relation between $a$ and $c$:

• Difference in age between $a$ and $c$: $|20 - 28| = |-8| = 8$.

Since $8 > 5$, the condition for $(a, c) \in R$ is not met. Thus, $(a, c) \notin R$.

We found a case where $(a, b) \in R$ and $(b, c) \in R$, but $(a, c) \notin R$. This violates the condition for transitivity.

Therefore, the relation is not transitive.

Summary of Properties

Based on our analysis, the relation "difference in ages is at most 5 years" on the set of persons in Delhi is:

• Reflexive: Yes
• Symmetric: Yes
• Transitive: No

This combination of properties matches one of the given options.

The relation is reflexive and symmetric but not transitive.