What Are Relations and Functions? – Class 12 Maths Chapter 1 Explained

If you are studying Class 12 Mathematics, Chapter 1 — Relations and Functions — is where your board exam preparation truly begins. This chapter builds on concepts introduced in Class 11 and deepens your understanding of how mathematical objects connect to each other. Whether you are preparing for CBSE board exams, CUET, or entrance tests like IPMAT, JIPMAT, or NPAT, mastering this chapter gives you a solid foundation.

What Is a Relation in Mathematics?

In everyday language, two objects are “related” if there is a recognisable connection between them. Mathematics formalises this using set theory.

A relation R from set A to set B is any subset of the Cartesian product A × B — a collection of ordered pairs (a, b) where a ∈ A and b ∈ B.

Real-World Examples

Let A be the set of Class XII students and B the set of Class XI students in the same school. Some meaningful relations from A to B include:

• {(a, b) ∈ A × B : a is a brother of b}

• {(a, b) ∈ A × B : age of a is greater than age of b}

• {(a, b) ∈ A × B : a lives in the same locality as b}

Not every student in A is connected to every student in B — the relation picks only those pairs satisfying the condition. We write a R b to indicate that (a, b) ∈ R.

What Is a Function in Mathematics?

A function is a special type of relation. Specifically, a function f : X → Y is a relation where every element of X maps to exactly one element of Y. The “exactly one” rule distinguishes functions from ordinary relations — two or more outputs for a single input means the rule is a relation but not a function.

Key Terminology

The distinction between co-domain and range is a common exam stumbling block. The range is what the function actually produces; the co-domain is what it is allowed to produce.

How Are Relations and Functions Connected?

Every function is a relation, but not every relation is a function. Think of relations as the broader category and functions as a stricter, more structured subset within it.

In Class 11 you encountered specific real-valued functions — polynomial, rational, modulus, signum, greatest integer — along with their graphs. Chapter 1 of Class 12 extends this to:

• Types of relations (reflexive, symmetric, transitive, equivalence)

• Types of functions (injective, surjective, bijective)

• Composition of functions and invertible functions

Each idea has direct applications in higher mathematics, engineering, and data science.

Why This Chapter Is Critical for Exams

Chapter 1 carries significant marks in the CBSE Class 12 board examination and forms a recurring topic in CUET (Mathematics), IPMAT, and NPAT.

Questions typically test:

• Classification of relations — is a given relation reflexive, symmetric, transitive, or an equivalence relation?

• Identification of function types — is a given function one-one, onto, or bijective?

• Composition and inverse — finding gof, fog, and f⁻¹.

Students who understand the definitions deeply, rather than memorising them, consistently perform better on these questions because the logic of each answer follows directly from the definition.

The Two Extreme Relations at a Glance

Before diving into detailed types, two extreme cases are worth knowing:

• Empty relation: R = φ. No element of A is related to any element. Example: On A = {1, 2, 3, 4}, R = {(a, b) : a − b = 10} has no valid pairs, so R is empty.

• Universal relation: R = A × A. Every element is related to every element. Both are called trivial relations.

Related Reading on ChampionsPrep

• Types of Relations: Empty, Universal, Reflexive, Symmetric and Transitive →

• What Is an Equivalence Relation? →

• One-One and Onto Functions Explained →

• Bijective and Invertible Functions →

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Test Your Knowledge

Q1.A relation R from set A to set B is defined as:

Any subset of A × BAny element of AA function from B to AThe union of A and B

Q2.What condition makes a relation a function f : X → Y?

Every element of Y is the image of two elements of XEvery element of X maps to exactly one element of YEvery element of X is related to every element of YX and Y must contain exactly the same number of elements

Q3.How are the range and the co-domain of a function related?

The co-domain is always a subset of the rangeThey are always equalThe range is always a subset of the co-domainThey have nothing in common

Q4.On A = {1, 2, 3, 4}, R = {(a, b) : a − b = 10} is:

The universal relationThe empty relationA reflexive relationA function

Q5.Which statement is correct?

Every relation is a function, but not every function is a relationFunctions and relations are two separate, unrelated ideasEvery function is a relation, but not every relation is a functionEvery function is a universal relation on its domain