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

Concept: Empty Relation and Universal Relation.

Step 1 – Empty relation:

$A$ is a set of students in a boys’ school, so every element of $A$ is male. The condition “$a$ is sister of $b$” can never be satisfied by any pair $(a,b)$ from $A$, because no student has a sister in the same set. Hence $R$ contains no ordered pairs.

Step 2 – Universal relation:

For any two students $a,b \in A$, the difference in their heights is at most a few meters. Since 3 meters is far greater than any possible height difference (maximum human height ~2.5 m), the condition “difference < 3 m” holds for every pair $(a,b)$ in $A \times A$. Thus $R' = A \times A$.

Concept first. A relation on a set $A$ is just a subset of $A \times A$.

• The empty relation has no pairs at all: $R = \varnothing$.
• The universal relation has every possible pair: $R = A \times A$.

The trick is to check whether the given condition can ever be satisfied, given the nature of $A$.

1. Why $R$ is empty.

   $A$ is the set of all students of a boys’ school. That means every element of $A$ is male.

   The condition for $(a,b) \in R$ is: “$a$ is the sister of $b$”.

   For $a$ to be a sister, $a$ must be female. But no element of $A$ is female.

   Hence there is no $a \in A$ that can satisfy the condition for any $b \in A$.

   So $R$ contains zero ordered pairs: $R = \varnothing$, which is the empty relation.

   Watch out

   A common mistake is to think “sister” implies a female outside the set. But the relation is defined on $A$ — both $a$ and $b$ must be from $A$. Since $A$ has only boys, $a$ can never be a sister.

2. Why $R'$ is universal.

   $R' = \{(a,b): \text{the difference between heights of } a \text{ and } b \text{ is less than } 3 \text{ metres}\}$.

   The maximum possible height of any human is well under 3 metres (the tallest recorded is about 2.72 m). So for any two students $a,b \in A$, the absolute difference in their heights is at most a few centimetres — certainly less than 3 metres.

   Therefore every ordered pair $(a,b) \in A \times A$ satisfies the condition.

   Hence $R' = A \times A$, which is the universal relation.

   Tip

   You don’t need exact heights. The key insight: 3 metres is larger than any possible height difference between two humans. So the condition is automatically true for all pairs — it’s a “vacuously satisfied” condition that makes the relation universal.

Method: Recognising the empty and the universal relation

Use this when a relation is described in words and you must show it is either empty (no pairs) or universal (all pairs of $A\times A$).

Steps

Step 1: Read the defining condition and the nature of the set

A relation on $A$ is a subset of $A\times A$. The question is: for which pairs $(a,b)$ from $A$ is the stated condition true?

Step 2: Test whether the condition can EVER hold

If the condition is impossible for every pair — because of what the elements of $A$ actually are — then no pair qualifies and the relation is empty ($R=\varnothing$).

Step 3: Test whether the condition ALWAYS holds

If the condition is guaranteed true for every pair (often because a bound in the condition is far larger than anything the elements can produce), then every pair qualifies and the relation is universal ($R=A\times A$).

The whole method is about reasoning from the meaning of the set's elements, not from listing pairs — you argue that the property is vacuously false or vacuously true.

Common Mistakes

Mistake 1: Thinking "sister" could be satisfied by someone outside the set

Why it's wrong: the relation is defined on $A$, so both $a$ and $b$ must be students of the boys' school; since none is female, no pair can satisfy "$a$ is sister of $b$". Correct approach: check the condition only for pairs drawn from $A$ itself — it fails for all, so $R$ is empty.

Mistake 2: Trying to prove $R'$ universal by measuring actual heights

Why it's wrong: exact heights are unknown and unnecessary, and chasing them wastes time. Correct approach: argue that $3$ metres exceeds any possible human height difference, so the condition holds for every pair automatically, making $R'=A\times A$.