Let $\mathrm{A}$ be the set of all students of a boys school. Show that the relation $\mathrm{R}$ in A given by $\mathrm{R}=\{(a, b): a$ is sister of $b\}$ is the empty relation and $\mathrm{R}^{\prime}=\{(a, b):$ the difference between heights of $a$ and $b$ is less than 3 meters $\}$ is the universal relation.

Step 1: Analyze the set A

The problem defines set A as the set of all students in a boys' school. This means that every element $a$ in set A is a boy. This fundamental characteristic of set A is crucial for analyzing the given relations.

Step 2: Analyze Relation R

For the relation R, an ordered pair $(a, b)$ belongs to R if $a$ is the sister of $b$. Since set A contains only boys, no student $a$ in A can be a sister to any other student $b$ in A. Therefore, there are no pairs $(a, b)$ that satisfy this condition.

Step 3: Conclude for Relation R

Since no pair $(a, b)$ from set A can satisfy the condition that $a$ is the sister of $b$, the relation R contains no elements. This means R is an empty relation.

Step 4: Analyze Relation R'

For the relation R', an ordered pair $(a, b)$ belongs to R' if the absolute difference between their heights is less than 3 meters. The height of any human being is typically between 0.5 meters (for a small child) and 2.5 meters (for a very tall adult).

Step 5: Evaluate height difference

The maximum possible difference between the heights of any two students would be less than 3 meters. For example, if one student is 0.5 meters tall and another is 2.5 meters tall, their difference is 2 meters, which is less than 3 meters. It is practically impossible for the height difference between any two students to be 3 meters or more.

Step 6: Conclude for Relation R'

Since the difference between the heights of any two students in the school will always be less than 3 meters, every possible ordered pair $(a, b)$ from set A will satisfy the condition for R'. Therefore, R' is the universal relation, which includes all possible pairs from A.

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