Relations and Functions

Notes

1. Relation

A relation R from set A to set B is a subset of A×B.

Example:

A = {1,2,3}, B = {4,5}

R = {(1,4), (2,5)}

2. Function

A function is a relation in which every element of the domain has exactly one image in the codomain.

Notation:

f:A→B

3. Domain

The set of all input values of a function.

4. Codomain

The set in which the outputs of a function lie.

5. Range

The set of actual output values of a function.

6. Types of Functions

One-One (Injective)

Onto (Surjective)

Bijective

Identity Function

Constant Function

7. Composition of Functions

(f∘g)(x)=f(g(x))

8. Inverse Function

Inverse exists only if the function is bijective.

Notation:

f

−1

(x)

9. Types of Relations

Reflexive

Symmetric

Transitive

Equivalence Relation

10. Identity Relation

I={(a,a):a∈A}