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

Question

vvLet \(L\) 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: Understanding Relation Definition

The relation \( R \) is given such that for any two lines \( \alpha \) and \( \beta \), \( \alpha R \beta \) if and only if \( \alpha \) is perpendicular to \( \beta \). Perpendicular means the lines intersect at a right angle.

Step 2: Testing Reflexivity

A relation is reflexive if for every element \( a \), \( aRa \) holds true. Applying this to the relation \( R \), reflexivity would mean every line \( \alpha \) is perpendicular to itself. However, no line is perpendicular to itself, so \( R \) is not reflexive.

Step 3: Testing Symmetry

A relation is symmetric if \( aRb \) implies \( bRa \). For the relation \( R \), if \( \alpha \) is perpendicular to \( \beta \), then \( \beta \) is also perpendicular to \( \alpha \). Hence, \( R \) is symmetric.

Step 4: Testing Transitivity

A relation is transitive if \( aRb \) and \( bRc \) imply \( aRc \). If \( \alpha \) is perpendicular to \( \beta \) and \( \beta \) is perpendicular to \( \gamma \), this does not necessarily mean that \( \alpha \) is perpendicular to \( \gamma \). Thus, \( R \) is not transitive.

Key Concepts

Reflexive Relation

A reflexive relation is a foundational concept in mathematics that deals with the idea of self-relatedness. For a relation to be reflexive, each element or entity must relate to itself. Let's unpack this further.

Consider a set, for instance, the set of all lines in a plane. A relation is reflexive if, for every line \(\alpha\) in this set, \(\alpha R \alpha\) holds true, meaning each line should satisfy the relation with itself. This property is quite intuitive in many mathematical structures.

However, in the context of our original exercise, which examines lines in a plane and the relation of being perpendicular, no line is perpendicular to itself because a perpendicular intersection implies a 90-degree angle, which cannot occur with a single line. Hence, this specific relation is not reflexive, illustrating how reflexivity depends heavily on the nature of the elements and the specific relation being examined.

Symmetric Relation

In mathematics, a symmetric relation is one in which if one element is related to another, then the second is also related back to the first. This symmetry signifies a reciprocal relationship between entities.

Let's visualize this with lines in a plane: if line \(\alpha\) is perpendicular to line \(\beta\), then, by symmetry, line \(\beta\) must automatically be perpendicular to line \(\alpha\). This is because perpendicularity inherently involves a mutual relationship, where the angle between the lines from either perspective is the same - precisely 90 degrees.

Therefore, the relation \(R\) described in the exercise is confirmed to be symmetric since the mutual perpendicular relationship holds true in both directions. Symmetric relations often help in scenarios where mutual interactions or reciprocal relationships need to be considered.

Transitive Relation

Transitive relations involve a chain-like logic where the relation from one to another can extend further. Specifically, if element \(a\) is related to element \(b\), and element \(b\) is in turn related to element \(c\), transitivity would imply that \(a\) must then be directly related to \(c\).

In our exercise, we examine if the perpendicular relation is transitive among lines. If \(\alpha\) is perpendicular to \(\beta\), and \(\beta\) to \(\gamma\), for transitivity, \(\alpha\) should also be perpendicular to \(\gamma\). However, this is not usually the case because two lines being perpendicular to the same line do not affect each other. They occupy distinct orientations relative to each other.

This limitation means our described relation \(R\) is not transitive, as the perpendicular link cannot be transferred through a common intermediary line. Recognizing non-transitive relations is crucial for understanding the limitations and boundaries of mathematical frameworks and structures.