Relations And Functions

 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 nm

|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 2mn .

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.

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