Function

 

Definition:   Given two sets X and Y,  a function  f  from X to Y is :

                     For any element in X,  it is mapped to one and only one element in Y.                  Usually, the mechanism is denoted as

 

                                            f: X   Y

 

                     The mapping is “injective”, namely, all the elements in X shall be mapped to Y.   The following diagrams would give you some sense of feeling about how it is defined.

 

 

           Example 1:

                                      

                                It is not a function from X to Y because the mapping for “a” is not defined.  However,  it is a function from the subset {b,c} to Y.

 

           Example 2:

                                  

                                    Yes, it is a function from X to Y.

 

            Example 3:

 

                                  

                                  No, it is not a function.  “c” is mapped to two elements in Y.

 

            Example 4:     Let X be the set for all positive numbers. 

 

                                  We define a mapping   for all elements in X such that

                                                   x  X,  the mapping     brings x  to  (x+3) .

 

                                  Then   is a function.  

                                  For simplicity,  we denote this mechanism as

 

                                                   :  x  X   x+3  ;

 

                                  or              (x) = x+3   .

 

                                                                       

 

            Example 5:    If x and y are with the following relationship as follows:

 

                                    

x	1	2	3	4
y	2	6	8	9

 

                                In other words, when x=1, y=2;

                                                          when x=2, y=6;

                                                         …

                                y varies as x varies. 

                       With this mapping, we can say y is a function of x.