“You must expect great things of yourself before you can do them.” –Michael Jordan

In this chapter, we studied different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows:

Empty relation is the relation R in X given by R = φ ⊂ X × X.

Universal relation is the relation R in X given by R = X × X.

Reflexive relation R in X is a relation with (a, a) ∈ R ∀a ∈ X.

Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.

Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.

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.

A function f : X → Y is one-one (or injective) if f(x_{1}) = f(x_{2} ) ⇒ x_{1} = x_{2} ∀ x_{1}, x_{2} ∈ X.

A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.

The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof(x) = g(f(x))∀ x ∈ A.

A function f : X → Y is invertible if ∃ g : Y → X such that gof = IX fog = IY

A function f : X → Y is invertible if and only if f is one-one and onto. and

Given a finite set X, a function f : X → X is one-one (respectively onto) if and only if f is onto (respectively one-one). This is the characteristic property of a finite set. This is not true for infinite set

A binary operation ∗ on a set A is a function ∗ from A × A to A. An element e ∈ X is the identity element for binary operation ∗ : X × X → X, if a ∗ e = a = e ∗ a ∀a ∈ X.

An element a ∈ X is invertible for binary operation ∗ : X × X → X, if there exists b ∈ X such that a ∗ b = e = b ∗ a where, e is the identity for the binary operation ∗. The element b is called inverse of a and is denoted by a^{–1}.

An operation ∗ on X is commutative if a ∗ b = b ∗ a ∀a, b in X.

An operation ∗ on X is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c)∀a, b, c in X.

## Select Lecture

- Empty Relations
- Universal Relations
- Trivial Relations
- Reflexive Relations
- Symmetric Relations
- Transitive Relations
- Equivalence Relations
- NCERT Exercise 1.1
- NCERT Examples
- Exemplar Problems
- Board’s Question Bank

- Reflexive Relations
- Symmetric Relations
- Transitive Relations
- Equivalence Relations
- Equivalence Classes
- Partition of Sets (Basics)
- NCERT Exercise 1.1
- NCERT Examples
- Exemplar Problems
- Board’s Question Bank

- One – One (Injective Functions)
- Many – One
- Onto (Surjective Functions)
- Into
- Bijective Functions
- NCERT Exercise 1.2
- NCERT Examples
- Board’s Question Bank

- One – One (Injective Functions)
- Many – One
- Onto (Surjective Functions)
- Into
- Bijective Functions
- NCERT Exercise 1.2
- NCERT Exemplar
- NCERT Examples
- Board’s Question Bank

- Composite Functions
- NCERT Exercise 1.3
- NCERT Examples
- Exemplar Problems
- Board’s Question Bank

- Invertible Functions
- NCERT Exercise 1.3
- NCERT Examples
- Exemplar Problems
- Board’s Question Bank