Function/Suggestion

Mathematical function

A mathematical function (also called mapping) is a means of associating every object in a certain set X with a unique object in a certain set Y. The set X is called the domain of the function, and the set Y is called the codomain of the function. The subset of Y that consists of all elements in Y that are associated with some elements in X is called the range of the function. The unique element in Y that is associated by a function f with x is usually written as f(x).

In Set Theory a function from a set X to a set Y is defined as a binary relation over X and Y that is both functional (i.e. associates with every element in X at most one element in Y) and total (i.e. associates with every element in X at least one element of Y). If the binary relation is only functional then it is called a partial function.

Functions can be specified in basically three ways. The simplest way is to simply enumerate the associated pairs. (Which is only possible if the domain is finite.) For example the function Weight that assigns to every living human being in the United States their weight in pounds, might be enumerated as follows.

• Larry -> 160, or Weight(Larry) = 160
• Jimmy -> 165, or Weight(Jimmy) = 165
• Ruth -> 125, or Weight(Ruth) = 125
• Cindy -> 120, or Weight(Cindy) = 120
• . . .

This sometimes also called specification by extension.

The second way of specifying a function is by giving an algorithm or an [algebraic expresssion]? that specifies how for every element in the domain the element in the codomain is determined. For example the following defines a function f:

f(x) = x⁵ - 1

(For a calculus of functions that are specified by algorithms see Lambda calculus.)

The third way of specifying a function is by giving an equation that relates variables that are associated with the domain and the codomain. The function f above, for example, can be specified with the following equation.

y = x⁵ - 1

The variable that is associated with the domain (here x) is called the independent variable and the variable associated with the codomain (here y) is called the dependent variable.

The last two ways of specifying a function are sometimes also called specification by intension.

Note that although originally the notion of function was primarily used to associate numbers with numbers, it can be used to associate any mathematical construct with any other mathematical construct such as vectors, matrices and all kinds of sets. An common example of this are vector valued functions that always result in a vector.

There are 4 basic kinds of functions. Let f be a function from X to Y then f is:

1. an into function if for some y in Y there is no x in X such that f(x) = y,
2. onto function or surjection if for every y in Y there is an x in X such that f(x) = y,
3. one-to-one function or injection if for every y in Y there is one and only one x in X such that f(x) = y, and
4. bijection if it is both "onto" and "one-to-one," or is both a surjection and an injection.

Examples:

• The function abs from Z (the set of Integers) to Z defined by

abs(x) = | x |

where | x | is the absolute value of x, is "into." Since there exists an integer -5 in Z such that there is no integer i in Z such that abs(i) = -5, abs is "into."

• Consider the function abs* from Z to Z* (the set of non-negative integers) that is defined by the same rule as abs. Then abs* is "onto," or a surjection, since for every y in Z*, that is, every non-negative integer, there is at least one x in Z such that abs*(x) = y.
• Consider the function add1 from Z to Z that is specified by the rule

add1(x) = x + 1

then add1 is "one-to-one" or an injection, since for every y in Z there is one and only one x in Z such that add1(x) = y.

• Consider the function 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 Z, there is only one x in Z such that add1(x) = y, that is, x = y - 1.