Question

Let \(\mathrm{H}\) be a set of hyperbolas. If a relation \(\mathrm{R}\) on \(\mathrm{H}\) be defined by \(\mathrm{R}=\left\\{\left(\mathrm{h}_{1}, \mathrm{~h}_{2}\right): \mathrm{h}_{1}, \mathrm{~h}_{2}\right.\) have same pair of asymptotes, \(\left.h_{1}, h_{2} \in H\right\\}\), then the relation \(R\) is (1) reflexive and symmetric but not transitive (2) symmetric and transitive but not reflexive (3) reflexive and transitive but not symmetric (4) Equivalence relation

Short answer

The relation \(\text{R}\) is an equivalence relation.

Step-by-step solution

Define the Relation

The relation \(\text{R}\) on the set of hyperbolas \(\text{H}\) is defined such that two hyperbolas \(\text{h}_1\) and \(\text{h}_2\) are related if they have the same pair of asymptotes. This is written as: \(\text{R} = \left\{ ( \text{h}_1, \text{h}_2) \: \: \text{h}_1, \text{h}_2 \text{ have the same pair of asymptotes} \right\}\).

Check Reflexivity

A relation \(\text{R}\) is reflexive if every element is related to itself. Since every hyperbola has the same pair of asymptotes as itself, \(\text{h} \text{R} \text{h}\) holds for any \(\text{h} \in \text{H}\). Thus, \(\text{R}\) is reflexive.

Check Symmetry

A relation \(\text{R}\) is symmetric if \(\text{h}_1 \text{R} \text{h}_2\) implies \(\text{h}_2 \text{R} \text{h}_1\). If \(\text{h}_1\) and \(\text{h}_2\) have the same pair of asymptotes, then \(\text{h}_2\) and \(\text{h}_1\) also have the same pair of asymptotes. Thus, \(\text{R}\) is symmetric.

Check Transitivity

A relation \(\text{R}\) is transitive if \(\text{h}_1 \text{R} \text{h}_2\) and \(\text{h}_2 \text{R} \text{h}_3\) imply \(\text{h}_1 \text{R} \text{h}_3\). If \(\text{h}_1\) and \(\text{h}_2\) have the same pair of asymptotes, and \(\text{h}_2\) and \(\text{h}_3\) have the same pair of asymptotes, then \(\text{h}_1\) and \(\text{h}_3\) must also have the same pair of asymptotes. Thus, \(\text{R}\) is transitive.

Conclude the Type of Relation

Since the relation \(\text{R}\) is reflexive, symmetric, and transitive, it satisfies all the properties of an equivalence relation.

Key concepts

reflexive relation

A reflexive relation on a set means that every element is related to itself. This property is essential for certain types of relations, like equivalence relations. To put it simply, if we have a relation \(\text{R}\) on a set \(\text{S}\), it is reflexive if for every element \(\text{s} \in \text{S}\), the pair \(\text{(s, s)}\) is in the relation \(\text{R}\).
Let’s consider our context of hyperbolas. Each hyperbola \(\text{h}\) will always have the same pair of asymptotes as itself. Therefore, for the given relation \(\text{R}\), which states that two hyperbolas \(\text{h}_1\) and \(\text{h}_2\) are related if they share the same pair of asymptotes, it’s easy to see that \(\text{h} \text{R} \text{h}\) for any hyperbola \(\text{h}\).
This means the relation is reflexive, as every hyperbola is related to itself.

symmetric relation

A symmetric relation is one where if one element is related to another, then the second element is also related back to the first.
In technical terms, for a relation \(\text{R}\) on a set \(\text{S}\), the relation is symmetric if for any elements \(\text{s}_1 \in \text{S}\) and \(\text{s}_2 \in \text{S}\), whenever \(\text{s}_1 \text{R} \text{s}_2\) is true, \(\text{s}_2 \text{R} \text{s}_1 \) must also be true.
When we apply this to our relation on hyperbolas, if we have \(\text{h}_1 \text{R} \text{h}_2\), this implies that \(\text{h}_1\) and \(\text{h}_2 \) have the same pair of asymptotes. Consequently, \(\text{h}_2\) and \(\text{h}_1\) must also have the same pair of asymptotes, hence \(\text{h}_2 \text{R} \text{h}_1\).
Therefore, the relation is symmetric.

transitive relation

A transitive relation means that if one element is related to a second, and the second is related to a third, then the first element must also be related to the third.
Mathematically, for a relation \(\text{R}\) on the set \(\text{S}\), it is transitive if whenever \(\text{s}_1 \text{R} \text{s}_2\) and \(\text{s}_2 \text{R} \text{s}_3\) hold true, then \(\text{s}_1 \text{R} \text{s}_3\) must also be true.
In the hyperbola context, if \(\text{h}_1 \text{R} \text{h}_2\) implies that \(\text{h}_1\) and \(\text{h}_2\) have the same pair of asymptotes, and if \(\text{h}_2 \text{R} \text{h}_3\) means that \(\text{h}_2\) and \(\text{h}_3\) share the same pair of asymptotes, then \(\text{h}_1\) and \(\text{h}_3\) must also have the same pair of asymptotes. Thus, \(\text{h}_1 \text{R} \text{h}_3\).
This confirms that the relation \(\text{R}\) is transitive.

equivalence relation

An equivalence relation is a special type of relation that combines the properties of reflexivity, symmetry, and transitivity.
In simpler terms, a relation \(\text{R}\) on a set \(\text{S}\) is considered an equivalence relation if it is:

• Reflexive: Each element is related to itself.
• Symmetric: If one element is related to another, then the second is related back to the first.
• Transitive: If one element is related to a second, and the second is related to a third, then the first element is related to the third.

Let’s apply this to hyperbolas. We have shown that the relation \(\text{R}\) on hyperbolas, defined by two hyperbolas having the same pair of asymptotes, is reflexive, symmetric, and transitive.
Because it satisfies all three properties, this makes \(\text{R}\) an equivalence relation.
In other words, it groups hyperbolas into classes where each class contains hyperbolas that share the same pair of asymptotes.