What is a Function ?

Function

Consider the relation  f : { (a,1),(b,2),(c,3),(d,5) }  from  set A = {a,b,c,d} to set B = {1,2,3,4}.

                                                

In this relation we see that each element of  A  has a unique image in  B . This relation  f  

from  set A  to set B  where every element of  A  has a unique image in  B  is defined as a

function from  A to B . So we observe that  in a function no two ordered pairs have

the same first element . 

We also see that ∃ an element  ∈ B  , i.e., 4 which does not have its preimage in  A . Thus

here: 

(i) the set B  will be termed as co-domain  and

(ii)  the set { 1,2,3,5 }  is called  range . From the above we can conclude that  range is a 

        subset of co-domain.

Symbolically, this function can be written as: