Problem 26 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(W\) denote the words in the English dictionary. Define the relation \(R\) by: \([2006]\) \(R=\\{(x, y) \in W \times W \mid\) the words \(x\) and \(y\) have at least one letter in common\\}. Then \(R\) is (A) not reflexive, symmetric and transitive (B) reflexive, symmetric and not transitive (C) reflexive, symmetric and transitive (D) reflexive, not symmetric and transitive

Step-by-Step Solution

Answer

(B) reflexive, symmetric and not transitive

Step 1: Understanding Reflexivity

A relation is reflexive if every element is related to itself. Here, for a word \( x \) in \( W \), \( (x, x) \) must be in \( R \) for the relation to be reflexive. Since every word has letters in common with itself, \( R \) is reflexive.

Step 2: Checking Symmetry

A relation is symmetric if whenever \( (x, y) \in R \), then \( (y, x) \in R \) as well. If two words \( x \) and \( y \) have a letter in common, then \( y \) and \( x \) also share that letter, making \( R \) symmetric.

Step 3: Evaluating Transitivity

A relation is transitive if whenever \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \in R \). For words \( x, y, \) and \( z \) to have transitive relationships, a shared letter between \( x \) and \( y \) and another between \( y \) and \( z \) doesn't guarantee \( x \) and \( z \) have a common letter, so \( R \) is not transitive.

Key Concepts

Symmetric Relation

In mathematical terms, a **symmetric relation** over a set is a relation where if one element is related to another, the reverse is also true. In this context of the provided exercise, we have a set of words where a relation \( R \) is defined based on the common letters between any two words. To determine symmetry for this relation, we consider any two words \( x \) and \( y \) within the set of words, \( W \).

• If \( x \) and \( y \) share at least one letter, represented as \((x, y) \in R\), then it's evident that \( y \) and \( x \) would also share that same letter.
• This symmetrical nature implies \( (y, x) \in R \) as well.

This mirrors a simple example: if "cat" and "hat" share the letter 'a', then both pairs, (cat, hat) and (hat, cat), naturally exist within the relation. Thus, the symmetry condition is satisfied, making \( R \) symmetric. Symmetric relations often make practical sense, as mutual sharing or interaction is common in various real-world contexts. This concept is widely applicable beyond words, such as in social networks where if person A is connected to person B, the connection often goes both ways.

Transitive Relation

A **transitive relation** relates to chaining a connection through an intermediate element. In the given problem, we analyzed a relation \( R \) concerning words. You'd typically think a relation might be transitive in a setting where a shared characteristic links more than two elements.

• For words \( x, y, \) and \( z \) in the set \( W \), if \( (x, y) \in R \) (\( x \) shares a letter with \( y \)), and \( (y, z) \in R \) (\( y \) shares a letter with \( z \)), logic might suggest that \( x \) and \( z \) should also share a letter for the relation to be transitive.

However, this scenario fails here because a shared letter between \( x \) and \( y \) and another between \( y \) and \( z \) doesn't ensure a direct link between \( x \) and \( z \). For example, the words "cat", "bat", and "bag" illustrate this. "Cat" and "bat" might share 'a', and "bat" and "bag" share 'b', but "cat" and "bag" do not have a common letter. Therefore, in this context, the relation \( R \) is **not transitive**.
Transitivity is a key property in many areas of mathematics but is context-sensitive, making it crucial to verify each case independently. Events in real-life where transitivity holds true include genealogy or hierarchies, such as if A is an ancestor of B, and B is an ancestor of C, then A is an ancestor of C.

Set Theory

Set theory is the study of collections of "things" or elements, where words like 'set', 'relation', and 'operation' are central. It provides the foundation for many concepts in mathematics, including how we define and work with relations.

• A **set** is a well-defined collection of distinct objects, considered as an object in its own right. The set \( W \) here refers to the collection of words in the English dictionary.
• A **relation** in set theory is a collection of ordered pairs, typically defined over Cartesian products.

In this exercise, the relation \( R \) is a specific subset of the product \( W \times W \), showcasing how relations can operate within sets. Set theory allows us to explore these relationships more formally, using basic principles to evaluate complex features like reflexivity, symmetry, and transitivity. By defining such properties, mathematicians can better understand the structural properties of collections, whether they are composed of words, numbers, or more abstract entities. This relates to broader mathematical disciplines where similar amendments in logical structure dictate operations and determine outcomes.