Class 12 Mathematics Chapter 1 Relations and Functions — Formulas & Key Points

Core Definitions — Relations

• Domain of R = {a ∈ A: (a, b) ∈ R for some b ∈ B}
• Range of R = {b ∈ B: (a, b) ∈ R for some a ∈ A}
• Codomain of R is the set B itself; range is always a subset of codomain

Properties of Relations — Quick Reference Table

• Reflexive: ∀ a ∈ A, (a,a) ∈ R — every element is related to itself
• Symmetric: (a,b) ∈ R ⇒ (b,a) ∈ R — relation works both ways
• Transitive: (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R — relation carries through a chain
• Equivalence: R is reflexive, symmetric and transitive all at once

Types of Functions — Definitions and Tests

• One-One (Injective): f(x₁)=f(x₂) ⇒ x₁=x₂, or equivalently x₁≠x₂ ⇒ f(x₁)≠f(x₂)
• Onto (Surjective): Range of f equals codomain B; every element in B is 'hit'
• Bijective: both one-one and onto; establishes a perfect pairing between A and B
• Many-One: at least two distinct elements in A map to the same element in B
• Into: at least one element in B has no pre-image in A

Composition of Functions — Formula and Properties

• (gof)(x) = g(f(x)); apply f first, then g
• Domain of gof ⊆ domain of f; specifically, x must satisfy f(x) ∈ domain of g
• Associative: ho(gof) = (hog)of
• Not commutative: usually gof ≠ fog
• Identity function I satisfies (Iof)(x)=f(x) and (foI)(x)=f(x)

Inverse Functions — Existence and Formula

• f invertible ⇔ f is bijective (one-one and onto)
• (fof⁻¹)(y)=y and (f⁻¹of)(x)=x (identity compositions)
• If f: A → B is invertible, then f⁻¹: B → A
• Graphs of f and f⁻¹ are reflections in the line y=x

Binary Operations — Definition and Properties

• Closure: a*b ∈ A for all a,b ∈ A
• Commutativity: a*b = b*a
• Associativity: (a*b)*c = a*(b*c)
• Identity e: a*e = e*a = a
• Inverse of a: a*a⁻¹ = a⁻¹*a = e

Common Mistakes and Notation Traps

• Codomain vs Range: codomain is the target set B; range is the actual set of output values, always ⊆ B
• One-One vs Onto: one-one concerns injectivity (no two inputs to same output); onto concerns surjectivity (every output is achieved)
• gof ≠ fog: composition order matters; always apply the rightmost function first
• f⁻¹(x) is the inverse function, not 1/f(x); write reciprocal as (f(x))⁻¹ or 1/f(x)
• Equivalence relation: must prove reflexive AND symmetric AND transitive; two out of three is not enough

Memory Tricks and Mnemonics

• RST = Reflexive, Symmetric, Transitive (all three for equivalence)
• Injective = Individual outputs; Surjective = Covers all of codomain
• gof → read right-to-left: 'f first, g second'
• CCIA checklist for binary operations: Closure, Commutativity, Identity, Associativity

Three Solved Mini-Examples Applying the Formulas

One-Glance Last-Minute Revision Box

• Relation on A: subset of A×A; Equivalence ⇔ Reflexive + Symmetric + Transitive
• Function types: One-One (f(x₁)=f(x₂)⇒x₁=x₂), Onto (range=codomain), Bijective (both)
• Composition: (gof)(x)=g(f(x)); associative, not commutative; domain condition range(f)⊆domain(g)
• Inverse: exists ⇔ f bijective; (fof⁻¹)=I and (f⁻¹of)=I; graph symmetry about y=x
• Binary operation *: Closure (a*b∈A), Commutativity (a*b=b*a), Identity e (a*e=a), Associativity ((a*b)*c=a*(b*c))
• Common pitfalls: confusing codomain/range, forgetting domain in composition, assuming commutativity, writing f⁻¹ for reciprocal