Class 12 Mathematics — Original Educational Content

Chapter 1: Relations and Functions 📐

Master the foundations of abstract mathematics — relations, equivalence classes, function types, composition, inverses, and binary operations.

📌 Relations & Equivalence | 🎯 Injective, Surjective, Bijective | 🧩 20 Practice Problems | 🎯 10-Question Quiz

Section 1

📌 Introduction to Relations

🔗 From Pairs to Patterns — The Language of Relationships

In everyday life, we constantly describe how things are connected: "Priya is taller than Ravi," "7 is greater than 3," "Mumbai is the capital of Maharashtra." Mathematics formalizes these connections using the powerful idea of a relation.

Before we can define a relation, we need to understand how to pair elements from two sets — enter the Cartesian Product! 🚀

Cartesian Product: Building All Possible Pairs

Given two non-empty sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.

A × B = {(a, b) : a ∈ A and b ∈ B}

Let A = {red, blue} and B = {1, 2, 3}. Then:

A × B = {(red,1), (red,2), (red,3), (blue,1), (blue,2), (blue,3)}

Notice: |A × B| = |A| × |B| = 2 × 3 = 6 ordered pairs.

Order matters! (red, 1) ≠ (1, red). The Cartesian product A × B is generally not the same as B × A.

What is a Relation?

A relation R from set A to set B is any subset of A × B. That is, R ⊆ A × B.

If (a, b) ∈ R, we write a R b and say "a is related to b under R."

A relation in a set A (or "on" A) means a relation from A to itself, i.e., R ⊆ A × A.

Let A = {1, 2, 3} and define a relation R by the rule "a is less than b." Then:

R = {(1,2), (1,3), (2,3)}

Here 1 R 2, 1 R 3, and 2 R 3 — but 3 R 1 is false (3 is not less than 1).

Domain, Codomain, and Range

For a relation R from A to B:

Domain of R = set of all first elements of the ordered pairs in R = {a : (a, b) ∈ R for some b ∈ B}
Codomain = the entire set B
Range of R = set of all second elements of the ordered pairs in R = {b : (a, b) ∈ R for some a ∈ A}

Range ⊆ Codomain, always. The range includes only those elements of B that are actually "hit" by R, while the codomain is the entire set B regardless of whether every element is used.

Two Extreme Relations

🕳️ Empty Relation (R = ∅)

No element of A is related to any element of A. R has zero ordered pairs.

Think of it as: "No student in the class has exactly 100 siblings" — the relation is empty because the condition is never satisfied.

🌐 Universal Relation (R = A × A)

Every element of A is related to every element of A (including itself). R contains ALL possible ordered pairs.

Think of it as: "Every student in the class breathes air" — the condition is satisfied by every pair, so all are related.

Both the empty relation ∅ and the universal relation A × A are trivially valid relations on A — they are subsets of A × A. Every other relation lies somewhere between these two extremes.

Section 2

🔄 Types of Relations

Not all relations behave the same way. We classify relations by three fundamental properties — each asking a simple question about the ordered pairs in R.

1. Reflexive Relations

A relation R on set A is reflexive if every element is related to itself:

∀ a ∈ A, (a, a) ∈ R

✅ Reflexive: R on ℤ defined by "a divides a." Since every integer divides itself (a = 1 × a), the pair (a, a) is always in R.

❌ Not Reflexive: R on ℝ defined by "a is strictly greater than b." We'd need a > a, which is impossible.

2. Symmetric Relations

A relation R on set A is symmetric if whenever a is related to b, then b is also related to a:

∀ a, b ∈ A, (a, b) ∈ R ⇒ (b, a) ∈ R

✅ Symmetric: R on a set of people defined by "a is a sibling of b." If Priya is Ravi's sibling, then Ravi is Priya's sibling.

❌ Not Symmetric: R on people defined by "a is the father of b." If Ramesh is the father of Suresh, Suresh is certainly not the father of Ramesh!

3. Transitive Relations

A relation R on set A is transitive if whenever a is related to b and b is related to c, then a is related to c:

∀ a, b, c ∈ A, (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R

✅ Transitive: "is less than" on ℝ. If a < b and b < c, then a < c.

❌ Not Transitive: "is perpendicular to" on lines in a plane. Line ℓ₁ ⊥ ℓ₂ and ℓ₂ ⊥ ℓ₃ does not imply ℓ₁ ⊥ ℓ₃ (in fact, ℓ₁ ∥ ℓ₃).

Quick Comparison Table

Relation (on appropriate set)	Reflexive	Symmetric	Transitive
"a = b" on ℤ	✓	✓	✓
"a < b" on ℝ	✗	✗	✓
"a ≤ b" on ℝ	✓	✗	✓
"a divides b" on ℤ⁺	✓	✗	✓
"a is sibling of b"	✗	✓	✓
"a ⊥ b" (lines)	✗	✓	✗

Consider the relation "is a friend of" on a social media platform (where you can be your own friend). Is it reflexive? If the platform auto-adds you as your own friend, yes. Symmetric? On Facebook, friendships are mutual — yes. Transitive? If A is friends with B and B is friends with C, is A necessarily friends with C? Not at all! So it's reflexive and symmetric, but not transitive.

A common error: "If R has no pairs of the form (a, b) and (b, c) where a ≠ c, then R is transitive." This is actually true — transitivity is satisfied vacuously! The condition "(a,b) ∈ R and (b,c) ∈ R" is never met, so the implication holds by default.