# Relations and Functions Notes, Class 12 Mathematics | SwaVid Relations and Functions, NCERT Class 12 Mathematics: notes with key ideas, formulas, common mistakes and 3 quick checks with worked answers. Canonical: https://www.swavid.com/cbse/class-12/maths/chapter/relations-and-functions Source: https://www.swavid.com/cbse/class-12/maths/chapter/relations-and-functions # Relations and Functions ## Formulas and definitions ## Key ideas ## Common mistakes ## Quick checks ## Keep going ## Questions parents ask ## Class 12 Mathematics on SwaVid ### Empty and universal relations ### Reflexive, symmetric, and transitive relations ### Equivalence relation and classes ### One-one or injective function ### Onto or surjective function ### Bijective function ### Next chapter ### Related chapter ### All chapters SwaVid › CBSE › Class 12 › Mathematics › Relations and Functions Notes written by SwaVid from the NCERT textbook. Class 12 Mathematics, Chapter 1. Key ideas, formulas and common mistakes. CBSE does not print marks for a single chapter. Short answer. This chapter studies different types of relations and functions on sets. Students learn to classify relations as reflexive, symmetric, transitive, or equivalence, and functions as one-one, onto, or bijective. Source: NCERT Mathematics Part I (Reprint 2026-27) , Chapter 1, Relations and Functions A relation R in a set A is called an empty relation if no element of A is related to any element of A, written as R = ∅. It is called a universal relation if each element of A is related to every element of A, so R = A × A. Both are called trivial relations. A relation R in A is reflexive if (a, a) ∈ R for every a ∈ A. It is symmetric if (a, b) ∈ R implies (b, a) ∈ R for all a, b ∈ A. It is transitive if (a, b) ∈ R and (b, c) ∈ R together imply (a, c) ∈ R for all a, b, c ∈ A. A relation R in set A is an equivalence relation if it is reflexive, symmetric, and transitive. An equivalence relation partitions A into mutually disjoint subsets called equivalence classes, where all elements within a class are related to each other and no element of one class is related to any element of another. A function f: X → Y is one-one or injective if the images of distinct elements of X under f are distinct. Formally, f(x₁) = f(x₂) must imply x₁ = x₂ for all x₁, x₂ ∈ X. A function that is not one-one is called many-one. A function f: X → Y is onto or surjective if every element of Y is the image of some element of X under f. In other words, for every y ∈ Y, there exists some x ∈ X such that f(x) = y. Equivalently, the range of f equals the codomain Y. A function f: X → Y is called bijective if it is both one-one and onto. A bijective function establishes an exact one-to-one correspondence between the domain X and the codomain Y. Try each question first. Then tap it to see the answer. Reflexive check: (1, 1), (2, 2), and (3, 3) are in R, so R is reflexive. Symmetric check: (1, 2) ∈ R and (2, 1) ∈ R, and all other pairs are of the form (a, a), so R is symmetric. Transitive check: (1, 2) ∈ R and (2, 1) ∈ R imply (1, 1) ∈ R, which is true; (2, 1) ∈ R and (1, 2) ∈ R imply (2, 2) ∈ R, which is true. Thus R is transitive. R is an equivalence relation One-one check: Let f(x₁) = f(x₂). Then 2x₁ + 3 = 2x₂ + 3, which gives 2x₁ = 2x₂, so x₁ = x₂. Thus f is one-one. Onto check: For any y ∈ R, set y = 2x + 3. Solving for x gives x = (y - 3) / 2, which belongs to R. Since f((y - 3) / 2) = y, f is onto. f is bijective For f to be onto, every element y ∈ N must satisfy f(x) = y for some x ∈ N. Take y = 1 ∈ N. Then 2x = 1 gives x = 1 / 2. Since 1 / 2 is not in N, the pre-image of 1 does not exist in N. f is not onto Inverse Trigonometric Functions . Continuity and Differentiability . The 13 chapters of Class 12 Mathematics . CBSE does not give marks to one chapter. It prints marks for units or parts, and the chapter list page shows them all. No. They are written by SwaVid from the NCERT textbook. The CBSE curriculum is the reference for what is assessed. Inverse Trigonometric Functions. SwaVid is an AI tutor for Indian students in Classes 6 to 12. It finds the earlier ideas a child missed, then teaches today’s chapter. Start free to see how it works. - Ch 1 NCERT book - 80 marks, theory paper - 3 quick checks - Assuming transitivity fails when no chain pairs exist . Transitivity only fails if you find pairs (a, b) ∈ R and (b, c) ∈ R such that (a, c) ∉ R. If no such pair with a matching intermediate element exists, the condition holds vacuously. - Confusing the codomain with the range of a function . The codomain is the entire given target set Y. The range is the set of actual outputs {f(x) : x ∈ X}. A function is onto only when its range equals the codomain. - Ignoring the domain when checking injectivity . Always check the domain of the function. For example, f(x) = x² is not one-one on the real numbers R because f(-1) = f(1), but it is one-one on the natural numbers N. ## Key Links - [CBSE](https://www.swavid.com/cbse) - [Class 12](https://www.swavid.com/cbse/class-12) - [Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup) - [All 13 chapters](https://www.swavid.com/cbse/class-12/maths/chapters) - [NCERT Mathematics Part I (Reprint 2026-27)](https://ncert.nic.in/textbook.php) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/inverse-trigonometric-functions) - [Open notes](https://www.swavid.com/cbse/class-12/maths/chapter/continuity-and-differentiability) - [Class 12 Mathematics](https://www.swavid.com/cbse/class-12/maths/chapters) - [Start free →](https://www.swavid.com/signup)