Chapter Summary: Relations and Functions

Key Concepts

• Relations: A relation R from set A to set B is a subset of A x B.
• Functions: A special type of relation where each element in the domain maps to exactly one element in the co-domain.

Types of Relations

• Empty Relation: R = Φ, no elements are related.
• Universal Relation: R = A x A, every element is related to every other element.
• Reflexive Relation: (a, a) ∈ R for all a ∈ A.
• Symmetric Relation: If (a, b) ∈ R, then (b, a) ∈ R.
• Transitive Relation: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R.
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

Functions

• One-One (Injective): f(x₁) = f(x₂) implies x₁ = x₂.
• Onto (Surjective): For every y in the co-domain, there exists an x in the domain such that f(x) = y.
• Bijective: A function that is both one-one and onto.

Important Definitions

• Equivalence Class: The subset of X containing all elements related to a.
• Composition of Functions: Combining two functions where the output of one function becomes the input of another.
• Invertible Functions: Functions that have an inverse.

Examples of Relations

1. Symmetric but not Reflexive or Transitive: R = {(1, 2), (2, 1)}.
2. Equivalence Relation: R = {(x, y): x and y have the same number of pages}.
3. Reflexive and Transitive but not Symmetric: R = {(a, b): a ≤ b}.

Common Pitfalls

• Confusing one-one and onto functions.
• Misidentifying types of relations (e.g., assuming a relation is equivalence without checking all properties).

Exam Tips

• Always verify the properties of relations and functions with examples.
• Practice identifying and constructing equivalence classes.

Chapter 1: Relations and Functions

1.1 Introduction

• The notion of relations and functions, domain, co-domain, and range were introduced in Class XI.
• A relation in mathematics is defined as a recognisable connection between two objects or quantities.
• Example of relations from set A (students of Class XII) to set B (students of Class XI):
  • (i) {(a, b) ∈ A × B: a is brother of b}
  • (ii) {(a, b) ∈ A × B: a is sister of b}
  • (iii) {(a, b) ∈ A × B: age of a is greater than age of b}
  • (iv) {(a, b) ∈ A × B: total marks obtained by a in the final examination is less than the total marks obtained by b}
  • (v) {(a, b) ∈ A × B: a lives in the same locality as b}
• A relation R from A to B is defined as an arbitrary subset of A × B.

1.2 Types of Relations

• Empty Relation: R = ∅ ⊆ A × A, where no element of A is related to any element of A.
• Universal Relation: R = A × A, where each element of A is related to every element of A.
• Reflexive Relation: R is reflexive if (a, a) ∈ R for every a ∈ A.
• Symmetric Relation: R is symmetric if (a₁, a₂) ∈ R implies (a₂, a₁) ∈ R for all a₁, a₂ ∈ A.
• Transitive Relation: R is transitive if (a₁, a₂) ∈ R and (a₂, a₃) ∈ R implies (a₁, a₃) ∈ R for all a₁, a₂, a₃ ∈ A.
• Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

1.3 Types of Functions

• One-One (Injective): A function f: X → Y is one-one if f(x₁) = f(x₂) implies x₁ = x₂.
• Onto (Surjective): A function f: X → Y is onto if for every y ∈ Y, there exists an x ∈ X such that f(x) = y.
• Bijective: A function is bijective if it is both one-one and onto.

Key Definitions and Properties

• Equivalence Class: [a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.
• Characteristic Property of Finite Sets: For a finite set X, a function f: X → X is one-one if and only if it is onto, and vice versa.

Historical Note

• The concept of function has evolved over time, starting from R. Descartes to G. H. Hardy.
• The modern set-theoretic definition of function was developed by Georg Cantor.

Examples of Relations

• Example of a relation that is symmetric but neither reflexive nor transitive: R = {(1, 2), (2, 1)}.
• Example of an equivalence relation: R = {(x, y) : x and y have the same number of pages in a library}.

Exercises

1. Determine the properties of various relations defined in different sets.
2. Show that specific functions are one-one, onto, or neither.

Common Mistakes and Exam Tips

Common Pitfalls

• Misunderstanding Relations: Students often confuse the definitions of reflexive, symmetric, and transitive relations. Ensure you understand each definition clearly:
  • Reflexive: For every element a in set A, (a, a) must be in relation R.
  • Symmetric: If (a₁, a₂) is in R, then (a₂, a₁) must also be in R.
  • Transitive: If (a₁, a₂) and (a₂, a₃) are in R, then (a₁, a₃) must also be in R.
• Equivalence Relations: Many students fail to check all three properties (reflexive, symmetric, transitive) when determining if a relation is an equivalence relation. Always verify all conditions.
• Function Types: Confusion often arises between one-one (injective), onto (surjective), and bijective functions. Remember:
  • One-one: Distinct elements in the domain map to distinct elements in the codomain.
  • Onto: Every element in the codomain is an image of some element in the domain.
  • Bijective: A function that is both one-one and onto.

Exam Tips

• Practice with Examples: Work through examples of each type of relation and function to solidify your understanding. For instance, show that the relation R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
• Use Diagrams: When studying functions, sketch diagrams to visualize mappings between sets. This can help clarify whether a function is one-one or onto.
• Check Extremes: For relations, remember the definitions of empty and universal relations. These can often be overlooked but are crucial for understanding the broader concepts.
• Review Definitions Regularly: Make flashcards for key definitions and properties of relations and functions to reinforce your memory.
• Solve Past Papers: Familiarize yourself with the format and types of questions that appear in exams by solving past papers.