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<x<=1000}. For example:

• Larry -> 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.

In terms of formal SetTheory, a Mapping from X to Y is usually defined as a MathematicalRelation where each x in X is related to one, and only one, element of Y. This element is the image of x. However, there are lots of other equivalent definitions.