Summary
NCERT Class 12 Maths Chapter 1 covers Relations and Functions, explaining types of relations (reflexive, symmetric, transitive, equivalence), types of functions (one-one, onto, bijective), composition of functions, and invertible functions.
Chapter 1 of NCERT Class 12 Mathematics (Part I) builds on Class 11 concepts to study types of relations and functions in depth. It defines empty, universal, reflexive, symmetric, transitive, and equivalence relations, showing how an equivalence relation partitions a set into disjoint equivalence classes. The chapter then classifies functions as one-one (injective), onto (surjective), or bijective, and introduces composition of functions and invertible functions, proving a function is invertible if and only if it is both one-one and onto.
Key points & formulas
- 01A relation R in set A is reflexive if (a, a) ∈ R for every a ∈ A, symmetric if (a, b) ∈ R implies (b, a) ∈ R, and transitive if (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R.
- 02An equivalence relation is one that is simultaneously reflexive, symmetric, and transitive; it partitions the set into mutually disjoint equivalence classes.
- 03A function f : X → Y is one-one (injective) if f(x₁) = f(x₂) implies x₁ = x₂, and onto (surjective) if every element of Y is the image of some element in X.
- 04A bijective function is both one-one and onto; for a finite set X, a function f : X → X is one-one if and only if it is onto — a property that does not hold for infinite sets.
- 05The composition gof of functions f : A → B and g : B → C is defined as gof(x) = g(f(x)); composition is not commutative in general (gof ≠ fog).
- 06A function f : X → Y is invertible if and only if it is bijective; its inverse g satisfies gof = I_X and fog = I_Y.
Frequently asked questions
01What is an equivalence relation in Class 12 Maths Chapter 1?
An equivalence relation R in a set A is a relation that is reflexive ((a, a) ∈ R for all a ∈ A), symmetric ((a, b) ∈ R implies (b, a) ∈ R), and transitive ((a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R). Such a relation divides the set into mutually disjoint subsets called equivalence classes.
02What is the difference between a one-one function and an onto function?
A one-one (injective) function f : X → Y maps distinct elements of X to distinct elements of Y — if f(x₁) = f(x₂) then x₁ = x₂. An onto (surjective) function ensures every element of Y is the image of at least one element in X, i.e., the range of f equals Y. A function that is both one-one and onto is called bijective.
03When is a function invertible according to NCERT Class 12 Chapter 1?
A function f : X → Y is invertible if there exists a function g : Y → X such that gof = I_X and fog = I_Y. The chapter proves that f is invertible if and only if it is both one-one and onto (bijective). The inverse function is denoted f⁻¹.
04Is the NCERT Class 12 Maths Chapter 1 PDF free to download?
Yes, the NCERT Class 12 Maths Chapter 1 PDF is completely free to download on cbseprepmaster.com.
More chapters in Mathematics Part I
This is the complete Mathematics Part I Chapter 1 as published by NCERT — every diagram, solved example, and exercise included, free. Browse all CBSE Class 12 textbooks.
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