Q.Show that the relation R in the set A of all the books in a library of a college, given by $R = \{(x, y) : x$ and $y$ have same number of pages$\}$ is an equivalence relation.

Concept: Relation Properties — To prove equivalence, we check reflexivity, symmetry, and transitivity.

Step 1: Reflexivity

Every book $x$ has the same number of pages as itself, so $(x, x) \in R$ for all $x \in A$. Hence $R$ is reflexive.

Step 2: Symmetry

If $(x, y) \in R$, then $x$ and $y$ have the same number of pages. Clearly $y$ and $x$ also have the same number of pages, so $(y, x) \in R$. Hence $R$ is symmetric.

Step 3: Transitivity …

Why this works — the core idea

An equivalence relation is just a way of saying "these things are the same in some specific way." Here, the "way" is having the same number of pages. The three properties — reflexivity, symmetry, transitivity — are the minimal checks to ensure that this "sameness" behaves consistently. If any one fails, the grouping breaks down.

Let’s verify each property step by step.

1. Reflexivity: Every book relates to itself

For any book $x$ in the library, the number of pages in $x$ is obviously equal to the number of pages in $x$. So $(x, x) \in R$ for every $x \in A$.

This is trivially true — no book can have a different page count from itself.

2. Symmetry: If $x$ relates to $y$, then $y$ relates to $x$

Take any two books $x$ and $y$ such that $(x, y) \in R$. By definition, this means $x$ and $y$ have the same number of pages.

But "same number" is a symmetric statement: if $x$ has the same page count as $y$, then $y$ automatically has the same page count as $x$. So $(y, x) \in R$ as well.

3. Transitivity: If $x$ relates to $y$ and $y$ relates to $z$, then $x$ relates to $z$ …

Method: Proving a "Same Value" Relation is an Equivalence Relation

Use this whenever a relation is "two objects share the same value of some attribute" (same page count, same age, same remainder). Such relations are always equivalence relations, and the proof follows a fixed template.

Steps

Step 1: Reflexive

Every object has the same attribute value as itself, so $(x,x)\in R$ for all $x$.

Step 2: Symmetric

"$x$ has the same value as $y$" is itself a symmetric statement, so it immediately gives "$y$ has the same value as $x$", i.e. $(y,x)\in R$.

Step 3: Transitive …

Common Mistakes

Mistake 1: Thinking related objects must be identical

Why it's wrong: "same number of pages" relates many distinct books; equality of the attribute is not equality of the objects. Correct approach: the relation groups objects by a shared value, not by being the same object.

Mistake 2: Skipping a property because it "obviously" holds

Why it's wrong: an equivalence-relation proof must state all three checks; examiners expect reflexive, symmetric, and transitive each addressed. Correct approach: write a line for each, even when brief. …