Class 12 Notes - Relations and Functions

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Relations and Functions

In this chapter, students will learn about the fundamental concepts of relations and functions, which are the building blocks of higher mathematics. Understanding these concepts is crucial for calculus, algebra, and many real-world applications.

1. Relations

  • Ordered Pair: A pair of elements written in a specific order, usually as (a, b).
  • Cartesian Product: For two sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
  • Relation: A relation R from set A to set B is a subset of the Cartesian product A × B. If (a, b) ∈ R, we say "a is related to b."
  • Types of Relations:
    • Reflexive Relation
    • Symmetric Relation
    • Transitive Relation
    • Equivalence Relation

2. Functions

  • Definition: A function f from set A to set B is a relation such that every element of A has a unique image in B. It is denoted as f: A → B.
  • Domain, Codomain, Range:
    • Domain: The set A (input values).
    • Codomain: The set B (possible output values).
    • Range: The set of actual output values.
  • Types of Functions:
    • One-one (Injective) Function
    • Onto (Surjective) Function
    • One-one and onto (Bijective) Function
    • Constant Function
    • Identity Function
    • Polynomial, Rational, Modulus, Signum, Greatest Integer Functions

3. Representation of Functions

  • Arrow diagrams
  • Set of ordered pairs
  • Algebraic formula
  • Graphical representation

4. Composition of Functions and Invertible Functions

  • Composition: If f: A → B and g: B → C, then the composition g ∘ f: A → C is defined by (g ∘ f)(x) = g(f(x)).
  • Invertible Function: A function f is invertible if there exists a function g such that g(f(x)) = x for all x in the domain of f.

5. Practice Problems

  1. Let A = {1, 2, 3}, B = {4, 5}. List all possible relations from A to B.
  2. Determine whether the relation R = {(1,1), (2,2), (3,3)} on A = {1,2,3} is reflexive, symmetric, and transitive.
  3. Give an example of a function that is one-one but not onto.
  4. If f(x) = x2 and g(x) = x + 1, find (g ∘ f)(2).
  5. Draw the graph of the greatest integer function.

6. Summary

  • Relations and functions connect elements of two sets in a specific way.
  • Functions are special types of relations with unique outputs for each input.
  • Understanding types and properties of relations and functions is essential for advanced mathematics.