** FUNCTIONS :
**

**Definition1: **A relation from a set A to a set B is called a FUNCTION , if every element of A is mapped with a unique element in B.it is denoted by f : A ? B.

A= Domain of f

B= Co-domain of f

**Definition2:** A function f : A ? B is a relation from A to B such that each a belongs to A belongs to a unique ordered pair(a,b) in f.

The number of functions that can be defined from A to B ,where |A|=m and |B|=n is n^{m}

|A|=number of elements in A.

**RELATIONS :
**

**Definition:**if A and B are two sets then every subset of Cartesian Product A×B is a relation from A to B.

The number of relations possible from A to B , where |A|=m and |B|=n is 2^{mn} .

## TYPES OF FUNCTIONS

**ONE-TO-ONE FUNCTION (INJECTION) : ** A Function f : A ? B is called an ONE-TO-ONE function , if distinct elements in domain A have distinct image in B.

**ON-TO FUNCTION (SURJECTION) :*** *A Function f : A ? B is called an ON-TO function , if each element of B is the image of atleast one element of A.

**ONE-TO-ONE AND ON-TO FUNCTION (BIJECTION):** A Function f : A ? B issaid to be a bijection if f is 1-1 and on-to.

**INVERSE OF A FUNCTION:** A Function f : A ? B is invertible if its inverse relation f^{ -1} is a function from B to A.

**CONSTANT FUNCTION:** f(x)=c for all x

**IDENTITY FUNCTION:** A function f : A ? A is called a Identity function if I = {(x,x) | x belongs to a}

## TYPES OF RELATIONS

**Reflexive Relation:** A Relation R on set A is called reflexive , if (x,x) belongs to R for all x belongs to A

example: if A = {1,2,3} then R= { (1,1) (2,2) (3,3) (1,2) (2,3) } is reflexive.

**Inverse Relation: **let R be a relation from set A to B . Then inverse of R, denoted by R ^{-1}_{ }= { (b,a) | (a,b) belongs to R }

example: if R= { (3,3) (1,2) (2,3) } then R ^{-1}_{ }={ (3,3) (2,1) (3,2) }

**Irreflexive Relation:** A Relation R on set A is called Irreflexive , if (x,x) does not belongs to R for all x belongs to A.

example: if A = {1,2,3} then R= { (1,2) (2,3) (3,2) (1,3) (2,1) } is irreflexive.

**Complement of a Relation:** let R be a relation from set A to B . Then Complement of R, denoted by R^{|}_{ }= (AxB) – R.

example: if A = {1,2,3} then R= { (1,2) (2,3) (3,2) (1,3) (2,1) } then R^{|} = { (1,1) (2,2) (3,1) (3,3) }

**Symmetric Relation: **A Relation R on set A is said to be Symmetric, if x R y then y R x for all x , y belongs to A.

example: R = { (1,1) (2,3) (3,2) } is Symmetric.

**Anti Symmetric Relation: **A Relation R on set A is said to be Anti Symmetric, if ( x R y and y R x ) then x=y for all x , y belongs to A.

example: A Relation <= (less than or equal to) is A nti Symmetric.

**Asymmetric Relation: **A Relation R on set A is said to be Asymmetric, if (x, y) belongs to R then (y,x) does not belongs to R, for all x , y belongs to A.

EXAMPLE: A relation < (less than ) is Asymmetric .

**Transitive Relation: ** A Relation R on set A is said to be Transitive, if ( x R y and y R z ) then x R z for all x , y,z belongs to A.

example: A relation <= is transitive on any set of real numbers.

**Equivalence Relation:** A Relation R on set A is said to be Equivalence , if R is( i) reflexive (ii) Symmetric (iii) Transitive

example: R = { (1,1) (2,2) (3,3) } is Equivalence.