Chapter 1 — Relations and Functions

Chapter 1 of the Class 12 Maths NCERT textbook, "Relations and Functions", covers types of relations (reflexive, symmetric, transitive, equivalence), types of functions (one-one, onto, bijective), composition of functions, and invertible functions.

• Classifying relations on a set — The chapter organises how elements of a set can be paired, moving from empty and universal relations up to reflexive, symmetric and transitive ones — and shows how a relation combining all three (an equivalence relation) neatly splits a set into disjoint classes.
• How functions map one set to another — Functions are studied by the way they cover their target set: injective functions never repeat an output, surjective ones reach every output, and bijective functions do both — a lens that later decides which functions can be reversed.
• Building and reversing functions — Two functions can be chained so the output of one feeds the next (composition), an operation that generally depends on order. The chapter ties this together by showing a function can be undone by an inverse precisely when it is a bijection.

1. 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.
2. 02An equivalence relation is one that is simultaneously reflexive, symmetric, and transitive; it partitions the set into mutually disjoint equivalence classes.
3. 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.
4. 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.
5. 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).
6. 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.