Problem 15 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(A) \(m \geq n\)

Step 1: Understanding Reflexive Relation

A reflexive relation on a set requires that every element in the set be related to itself. This means that for a set with \(n\) elements, there are \(n\) fundamental ordered pairs \((a, a)\) where \(a\) belongs to the set.

Step 2: Counting Required Pairs

Since each element must be related to itself for the relation to be reflexive, the minimal number of pairs in the relation \(R\) must be \(n\), i.e., one pair for each element.

Step 3: Counting Additional Pairs

The total number of pairs \(m\) in \(R\) includes at least the \(n\) reflexive pairs. There might be additional pairs as R can have other pairs besides the reflexive ones, increasing \(m\) beyond \(n\).

Step 4: Concluding the Minimum Number

Since every reflexive relation must at least include the \(n\) pairs where every element is related to itself, \(m\) has to be at least \(n\). Hence, \(m \geq n\).

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with collections of objects, or "sets." A set can be comprised of anything: numbers, letters, other sets, etc. The concept of a set is intuitive and simple: it's a group of things collected together.
You can denote a set by listing its elements. For example, if you have a set consisting of numbers 1, 2, and 3, you can write this set as \( A = \{1, 2, 3\} \).
Sets are often used to describe mathematical concepts, including relations and functions. In set theory, it's essential to understand terminology like elements (or members) of a set and subsets (a set contained within another set).

• An element belongs to a set if it is listed as one of the set's members.
• A subset is a set whose elements are all contained in another set. If \( B \subseteq A \), every element of \( B \) is also an element of \( A \).

Understanding these basic components of set theory allows you to tackle more complex topics, such as reflexive relations on sets.

Ordered Pairs

Ordered pairs are a way to keep track of two elements arranged in a specific order. An ordered pair is written as \((a, b)\), where the order in which \(a\) and \(b\) appear is crucial.
For instance, \((2, 3)\) is different from \((3, 2)\). This concept is used heavily in defining relations in mathematics. Ordered pairs are the building blocks of relations and functions.
In the context of set theory and relations, each pair represents a relationship between two elements of a set. If you have a set \(A\), an ordered pair \((a, b)\) means that there is some form of relationship between \(a\) and \(b\) that is defined by a particular relation.

• Ordered pairs are used to map one element to another, often to define a relation like equality, inequality, or a specific property.
• When sets of ordered pairs are collected, they form what is called a "relation." This collection defines how each element in the set is related to others.

Grasping the concept of ordered pairs is vital for understanding how relations like reflexive, symmetric, and transitive are structured.

Relation Properties

In set theory, a relation on a set is a collection of ordered pairs. These pairs describe how elements of the set relate to each other. To understand the different types of relations, it's important to learn about relation properties like reflexivity, symmetry, and transitivity.
A **reflexive relation** is one in which every element of a set is related to itself. For a set \(A = \{a_1, a_2, \, ..., \, a_n\}\), the relation \( R \) is reflexive if and only if \((a_i, a_i)\) is included in \( R \) for every \(i\).
This implies that at minimum, a reflexive relation on a set with \(n\) elements must contain at least \(n\) ordered pairs, all of which are of the form \((a_i, a_i)\). Hence, the statement \(m \geq n\) in the exercise solution.

• A reflexive relation can have additional pairs beyond these \(n\) pairs and still remain reflexive. These additional pairs increase \(m\) beyond \(n\), but they are not required to maintain reflexivity.
• Learning to identify a reflexive relation involves ensuring every element in the set has a pair with itself within the relation.

Understanding relation properties helps in determining whether a relation like \(R\) maintains certain characteristics such as reflexivity, which is a fundamental aspect in numerous areas of mathematics.