MappinG

A MappinG is simply a “rule” that assigns to each member of a Set A, a unique element of a set B.

There are non-mathematical MappinGs. Consider the “rule,” WGT that assigns to every living human being in United States their weight in pounds.  Then the set A = {people living in the United States} and B = {x:  0 160 or WGT (Larry) = 160
*                       Jimmy  -> 165 or WGT (Jimmy) = 165
*                       Ruth   -> 125 or WGT (Ruth)  = 125
*                       Cindy  -> 120 or WGT (Cindy) = 120
*                         .
*                         .
There are mathematical MappinGs as well. Consider the “rule,” ABS that assigns to each integer, its absolute value. Let set C = I, and the set D = I, also. Then, for example:
*                     -5         ->  5 or ABS (-5) = 5
*                      5         ->  5 or ABS (5)  = 5
*                     19         -> 19 or ABS (19) = 19
*                   -145         -> 14 or ABS (-145) = 145
*                        .
*                        .


There are 4 basic kinds of MappinGs.

1) ''into'' MappinG: this is a MappinG from a set X to a set Y such that there exists a y in Y such that there is no x in X such that x is mapped to y. 


2) ''onto'' MappinG: this is MappinG from a set X to a set Y such that for every y in Y there is at least one x in X such that x is mapped to y. Such a MappinG is called a SurJection.


3) ''one-to-one'' MappinG : this is a MappinG from a set X to a set Y such that for every y in Y there is one and only one x in X such that x is mapped to y. Such a mapping is called an InJection.
 
4) Further, a MappinG that is both "onto" and "one-to-one," or is both a SurJection and an InJection is called a BiJection.

Examples:

* The mapping defined above as ABS, defined from the set I = Integers to the set I = integers is "into." Since the exists an integer –5 in I such that there is no integer i in I such that ABS (i) = -5, ABS defined from I -> I is "into."

* Consider the MappinG, ABS, defined from I = IntegerNumbers to I* = { IntegerNumbers >= 0}. Then ABS defined from I to I* is "onto," or a SurJection, since for every y in I*, that is, every positive integer, there is at least one x in  I such that ABS (x) = y.

* Consider the MappinG, ADD1, defined from I to I such that if x is in I then ADD1 (x) = x+1. Then ADD1 is "one-to-one" or an InJection, since for every Y in I there is one and only one x in I such that ADD1 (x) = y.

* Consider the MappinG, ADD1, as defined above. Then ADD1 is both "onto" and "one-to-one" or a BiJection, since it is already “onto” and we can show it is "one-to-one" as well. Thus, given a y in I, there is only one x in I such that ADD1 (x) = y, that is, x = y-1.