Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(L\) denotes the set of all straight lines in a plane. Let a relation \(R\) be defined by \(\alpha R \beta \Leftrightarrow \alpha \perp \beta, \alpha, \beta \in L\) Then \(R\) is (A) reflexive (B) symmetric (C) transitive (D) None of these

Step-by-Step Solution

Answer

(B) symmetric

Step 1: Understand Reflexive Property

A relation is reflexive if every element is related to itself. In this context, for a line \( \alpha \), if \( R \) were reflexive, then \( \alpha \perp \alpha \). However, a line cannot be perpendicular to itself, so \( R \) is not reflexive.

Step 2: Check Symmetric Property

A relation is symmetric if whenever \( \alpha R \beta \), then \( \beta R \alpha \). Here, if line \( \alpha \perp \beta \), then it is also true that \( \beta \perp \alpha \). Therefore, the relation \( R \) is symmetric.

Step 3: Explore Transitive Property

A relation is transitive if whenever \( \alpha R \beta \) and \( \beta R \gamma \), then \( \alpha R \gamma \). Since \( \beta \perp \alpha \) and \( \beta \perp \gamma \) doesn't guarantee \( \alpha \perp \gamma \), \( R \) is not transitive.

Key Concepts

Reflexive Property

In mathematics, a relation is termed reflexive if each element in a set is related to itself under the given relation. To understand this better, consider the set of all straight lines \( L \) in a plane. If we have a relation \( R \) defined by \( \alpha R \beta \Leftrightarrow \alpha \perp \beta \), it indicates that for the relation to be reflexive, every line \( \alpha \) should be perpendicular to itself, meaning \( \alpha \perp \alpha \). However, a line cannot be perpendicular to itself, as per basic geometric principles. This means that the relation \( R \) in this particular context is not reflexive because it does not satisfy the reflexive property condition.

• In general, reflexivity requires self-relationship.
• Lines cannot be perpendicular to themselves; hence the property fails here.

Understanding reflexivity is crucial as it helps identify whether a set behaves consistently with this property across its elements.

Symmetric Property

The symmetric property in mathematics indicates that if one element is related to another, then the reverse is also true. It revolves around mutual relation between pairwise elements. Let's consider our exercise in which a relation \( R \) is defined such that a line \( \alpha \) is perpendicular to a line \( \beta \). The question is, "If \( \alpha \perp \beta \), is it always true that \( \beta \perp \alpha \)?"
The answer is yes, because the perpendicularity between two lines is inherently a mutual relationship. Thus, if line \( \alpha \) is perpendicular to line \( \beta \), then line \( \beta \) is also perpendicular to line \( \alpha \).

• This mutual nature confirms that the relation \( R \) is symmetric in this context.
• Symmetric relations allow interchangeability of elements without loss of relational validity.

Understanding this property helps in recognizing one-to-one correspondence in relations where symmetry holds. It is a crucial aspect of defining relationships between set elements, as seen in our line example.

Transitive Property

A relation is regarded as transitive if it carries through a chain of relations among three or more elements in the set. Let's dive into the original problem where the relation \( \alpha R \beta \Leftrightarrow \alpha \perp \beta \) is defined, and consider whether this relation is transitive.
To be transitive, whenever line \( \alpha \perp \beta \) and \( \beta \perp \gamma \), it must also follow that \( \alpha \perp \gamma \). However, in the context of perpendicularity, this condition doesn't always hold. Just because line \( \alpha \) is perpendicular to line \( \beta \) and \( \beta \) is perpendicular to line \( \gamma \), it does not logically imply that \( \alpha \) is perpendicular to \( \gamma \).

• Transitivity requires a chain reaction of relational properties across elements.
• Perpendicularity does not inherently propagate through a sequence of lines.

Thus, the relation \( R \) fails to meet the transitive property requirements, highlighting the importance of this property in verifying the logical flow of relations across elements in a set.