Function

FUNCTION

Definition of function

function is a relationship between two quantities, one of which is completely determined by the value of the other. A function f from a set X to a set Y is a relation between the elements of X (called the inputs) and the elements of Y (called the outputs) with the property that each input is related to one and only one output. We use the notation

f : X → Y

Concept of Boolean Functions

An expression that is form with binary variables. It can be represented as an algebraic expression or in truth table.

      

       TYPES  OF FUNCTION

1.      INJECTIVE

o   one-to-one

o   f  is called injective when the a = b

o   can express in quantifier

2.      SURJECTIVE

o   onto

o   f  is called surjective when there is a function from element A to element B

3.      BIJECTIVE

o   one-to-one correspondence

o   f  becomes function when no value in  the domain are signed to the same function value

o   no repeatation of domain

Inverse function and composition of function

Inverse function

Definition :-

¨      Let f  be one-to-one correspondence from the set A to the set B

¨      f  is the function that assign to be an element b belonging to B the unique element a in A such that f (a) = b

¨      Function of f is denoted by  f ˉ ¹ then  f ˉ ¹( b) = a when f (a) = b

       

        example

                           is the relation 

      

Composition function

¨       Let g be a function from the set A to the set B and let f be a function from the set B to the set C

¨      Function f and g, denoted for all a Î A by  f  ° g is defined by

      

           ( f  ° g )(a) = f( g(a) )

      

  example

                   f  ° g is defined  by   ( f  ° g )(a) = f  ( g(a)) = f (b) = 2

               ( f  ° g )(b) =  f ( g(b)) = f(c) =1 and ( f  ° g )(c) =  f ( g(c)) = f(a) =3

       Noted that:-

                f  ° g is not defined because the range of   f  is not a subset of domain of  g