Problem 15 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(R\) be a reflexive relation on a finite set \(A\) having \(n\) elements, and let there be \(m\) ordered pairs in \(R\). Then (A) \(m \geq n\) (B) \(m \leq n\) (C) \(m=n\) (D) None of these

Step-by-Step Solution

Answer

The correct choice is (A) \( m \geq n \).

Step 1: Understanding Reflexive Relation

A relation \( R \) on a set \( A \) is reflexive if, for every element \( a \) in \( A \), the pair \( (a, a) \) is in \( R \). This means that for a set \( A \) with \( n \) elements, there are at least \( n \) pairs \( (a_i, a_i) \) in the relation \( R \).

Step 2: Counting Elements in Reflexive Relation

Since the relation is reflexive, there are at least \( n \) pairs of the form \( (a_i, a_i) \). Therefore, the minimum number of ordered pairs \( m \) in \( R \) is \( n \). This tells us \( m \geq n \).

Step 3: Evaluating Option Choices

Now, let's consider the options: \( m = n \) is possible if \( R \) contains only the reflexive pairs. However, \( m \) can be greater than \( n \) if \( R \) includes more pairs beyond just the reflexive ones, which makes \( m \geq n \) the guaranteed condition for any reflexive relation. Thus, the correct choice is \( (A) \ m \geq n \).

Key Concepts

Finite Set

A finite set is a collection of distinct elements that has a specific, countable number of members. For instance, the set \( A = \{1, 2, 3\} \) is a finite set because it contains exactly three elements.
In the context of the exercise, we are interested in a finite set \( A \) which consists of \( n \) elements. Each set member can be paired with itself and potentially with other members of the set when forming relations. The concept of a finite set is crucial in determining the limits and characteristics of relations, especially reflexive ones, where each element needs to relate to itself.

Finite sets are helpful because they make it practical to count ordered pairs and manage the relations within the set. The finite nature implies constraints such as a known number of pairs in any specific kind of relation, such as reflexivity. Understanding finite sets allows for an organized approach to examining properties and behaviors of relations.

Ordered Pairs

Ordered pairs are a fundamental component of relations in set theory. An ordered pair is formed by combining two elements in a specific sequence, denoted by \((a, b)\). The order matters here, meaning \((a, b)\) is not the same as \((b, a)\) unless \(a = b\).
In the exercise, we focus on ordered pairs that form the reflexive relation \( R \) on a finite set \( A \). Each element \( a \) of a set with \( n \) elements will possess a reflexive ordered pair, i.e., \((a, a)\).

This requirement ensures that for a reflexive relation, at least \( n \) reflexive pairs must exist in the set \( R \), one for each element. Additionally, ordered pairs can include combinations that relate an element to others beyond its self-pairing in more extensive relations. Recognizing the significance of each pairing and sequence aids in clearly distinguishing what forms a part of \( R \), impacting how we calculate the number of pairs \( m \).

Relation Properties

Relation properties help us understand how elements of a set interact with each other within the context of a relation. There are several key properties, but here we emphasize reflexivity, a crucial aspect of this exercise.
A relation \( R \) on a set \( A \) is reflexive if every element \( a \) in \( A \) satisfies \((a, a) \in R\). Reflexive properties ensure that every element relates to itself, establishing a basic requirement for ordering pairs. This characteristic plays a pivotal role in calculating the minimum size of \( m \), which denotes the total number of ordered pairs.

The reflexive nature guarantees a baseline of \( n \) self-paired elements in \( R \). Beyond reflexivity, other types of relations, such as symmetric or transitive, may influence the structure but aren't a compulsory part of \( R \) being reflexive. Understanding these relation properties is vital for determining how elements and sets are interconnected, forming the basis of many mathematical concepts and solutions.