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Relations and Functions

Punjab Board · Class 12 · Mathematics

Quick revision notes for Relations and Functions — Punjab Board Class 12 Mathematics. Key concepts, formulas, and definitions for last-minute revision.

45 questions25 flashcards5 concepts

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A visual representation of a relation R from set A to set B as a subset of the Cartesian product A x B, showing elements of A mapped to elements of B.
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Key Topics to Revise

1

Types of Relations

  • A relation R in a set A is a subset of A × A (the Cartesian product of A with itself).
  • Empty Relation: R = φ (no element is related to any other element). Example: R = {(a,b): a – b = 10} in A = {1,2,3,4} is empty.
  • Universal Relation: R = A × A (every element is related to every element). Example: R = {(a,b): |a – b| ≥ 0} in A = {1,2,3,4} is universal.
2

Types of Functions

  • One-One (Injective) Function: Different inputs always give different outputs. f(x₁) = f(x₂) ⟹ x₁ = x₂.
  • Many-One Function: Two or more different inputs can give the same output. Example: f(x) = x² is many-one since f(1) = f(–1) = 1.
  • Onto (Surjective) Function: Every element of the co-domain Y is the image of at least one element in the domain X. Range of f = Y.
3

Composition of Functions

  • If f: A → B and g: B → C, then the composition gof: A → C is defined as gof(x) = g(f(x)) for all x ∈ A.
  • The composition 'gof' means 'first apply f, then apply g'. Read from right to left.
  • gof ≠ fog in general — composition is NOT commutative.
4

Invertible Functions

  • A function f: X → Y is invertible if there exists a function g: Y → X such that gof = Iₓ and fog = Iy.
  • g is called the inverse of f and is written as f⁻¹.
  • CRITICAL: A function is invertible if and only if it is BIJECTIVE (both one-one and onto).

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Full Notes

Key Concepts

A relation R from setA relation R in setA relation R in setA relation R in setA relation R in set

Frequently Asked Questions

What are the important topics in Relations and Functions for Punjab Board Class 12 Mathematics?
Key topics in Relations and Functions include Complete Overview of Relations and Functions Concepts, Diagram showing how a relation connects elements from Set A to Set B, Decision tree showing classification of relations as empty, universal, or partial. These are the concepts Punjab Board Class 12 examiners draw on most — study them first, then practise related questions.
How to score full marks in Relations and Functions — Punjab Board Class 12 Mathematics?
Understand the core concepts first, then work through the 45 practice questions available for this chapter. Revise formulas and definitions regularly, and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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