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## Kerala Plus Two Maths Notes Chapter 1 Relations and Functions

Introduction

A relation from a non-empty set A to a non-empty set B is a subset of A × B. In this chapter we study different types of relations and functions, composition of functions, and binary operations.

Basic Concepts

I. Types of Relations:

Here we study different relations in a set A

Empty Relation:

R : A → A given by R = Φ ⊂ A × A.

Universal Relation:

R : A → A given by R = A × A.

Reflexive Relation:

R : A → A with (a, a) ∈ R, ∀a ∈ A.

Symmetric Relation:

R : A → A with

(a, b) ∈ R ⇒ (b, a) ∈ R, ∀a, b ∈ A.

Transitive Relation:

R : A → A with (a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R,

∀a, b, c ∈ A.

Equivalence Relation:

R : A → A which is Reflexive, Symmetric, and T ransitive.

Equivalence Class:

Let R be an Equivalence Relation in a set A. If a ∈ A, then the subset {x ∈ A, (x, a) ∈ R} of A is called the Equivalence Class corresponding to ‘a’ and it is denoted [a].

II. Types of Functions

One-One or Injective function.

A function f : A → B is said to be One-One or Injective, if the image of distinct elements of A under fare distinct.

i.e; f(x_{1}) = f(x_{2}) ⇒ x_{1} = x_{2}

Otherwise f is a Many-One function.

1. Graphical approach:

If lines parallel to x-axis meet the curve at two or more points, then the function is not one-one.

Onto or Surjective function:

A function f : A → B is said to be Onto or Surjective, if every element of B is some image of some elements of A under f.

ie; If for every element y ∈ Y then there exists an element x in A such that f(x) = y.

Bijective function:

A function f : A → B is said to be Bijectiveit it is both One-One and Onto.

Composition of Functions.

Let f : A → B and g : B → C be two functions. Then the composition of f and g denoted by is gof defined

gof : A → C and gof (x) = g(f(x)).

- If f : A → B and g : B → C are One-One, then gof : A → C is One-One.
- If f : A → B and g : B → C are Onto, then gof : A → C is Onto.
- If f : A → B and g : B → C are Bijective, ⇔ gof : A → C is Bijective.

Inverse Function:

If f : A → B is defined to be invertible, if there exists a function g : B → A such that gof = I_{A} and

fog = I_{B}. The function g is called the inverse of ‘f’ and is denoted by f^{-1}.

- If function f : A → B is invertible only if f is bijective.
- (gof)
^{-1}= f^{-1}og^{-1}.

III. Binary Operations

A binary operation ‘*’ on a set A is a function * : A × A → A, defined by a * b, a, b ∈ A.

- * : A × A → A is commutative if a * b = b * a, ∀a, b ∈ A.
- * : A × A → A is associative if a * (b * c) = (a * b) * c, ∀a, b, c ∈ A.
- e ∈ A is the identity element for the binary operation * : A × A → A if a * e = a = e * a, ∀a ∈ A.
- An elements a ∈ A is invertible for the binary operation * : A × A → A, if there exists an element b ∈ A such that a * b = e = b * a, where ‘e’ is the identity element for the operation ‘*’. Then ‘b’ is denoted by a
^{-1}.