Relations and Functions – Class 11 Maths (NCERT Based)
The chapter Relations and Functions is a fundamental topic in Class 11 Mathematics. It introduces important concepts that are used throughout higher mathematics, including calculus, probability, and statistics. A clear understanding of this chapter is essential for success in later chapters and competitive exams.
This content is prepared strictly according to the NCERT Class 11 Maths syllabus and explained in a simple, student-friendly manner.
📌 What is a Relation?
A relation is a connection between elements of two sets.
If A and B are two non-empty sets, then a relation from A to B is a subset of the Cartesian product A × B.
Example:
If A = {1, 2} and B = {3, 4}, then
A × B = {(1,3), (1,4), (2,3), (2,4)}
Any subset of this set is a relation.
✍️ Types of Relations
According to NCERT, relations are classified as:
- Empty Relation – No element is related
- Universal Relation – All possible ordered pairs
- Identity Relation – Each element is related to itself
- Reflexive Relation – Every element is related to itself
- Symmetric Relation – If (a, b) then (b, a)
- Transitive Relation – If (a, b) and (b, c), then (a, c)
- Equivalence Relation – Reflexive, symmetric, and transitive
📖 What is a Function?
A function is a special type of relation where each element of the first set is related to exactly one element of the second set.
If f is a function from set A to set B, it is written as:
f : A → B
🔢 Domain, Codomain, and Range
- Domain – Set of all input values
- Codomain – Set of all possible outputs
- Range – Actual outputs of the function
Understanding these three terms is very important for solving function-based problems.
📐 Types of Functions (NCERT)
- One-One Function (Injective) – Different inputs give different outputs
- Many-One Function – Different inputs give the same output
- Onto Function (Surjective) – Every element of codomain has an image
- Into Function – Some elements of codomain have no image
- One-One and Onto (Bijective) – Both injective and surjective
🔁 Composition of Functions
If f and g are two functions, then their composition is written as:
(g ∘ f)(x) = g(f(x))
Composition of functions is an important concept used in advanced mathematics.
🔄 Invertible Functions
A function is invertible if its inverse exists.
Only one-one and onto functions have inverse functions