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Posted 15 April 2008

EXAMPLES OF FUNCTIONS

Basic functions

The examples given here:

¨  are basic types of functions that are used everywhere in abstract math

¨  are functions you need to be familiar with

¨  are (sigh) boring.

Identity function

For any set A, the identity function  is the function that takes an element to itself; in other words, for every element , . Thus its graph is the diagonal of . 

Usage

¨ Warning:  The word identity has two other commonly used meanings.  This causes trouble because people may refer to the identity function as simply “the identity”, especially in conversation.

¨ The notation  for the identity function on A is fairly common, but  is also very common. 

Properties

The identity function is injective and surjective.

Understanding the identity function

¨  The identity function on a set A is the function that does nothing to each element of A.

¨  The identity function on  is the familiar function defined by .  Its graph in the plane is the diagonal  line from lower left to upper right through the origin.  Its derivative is the constant function defined by g(x) = 1.

¨  There is a different identity function for each different set.  See overloaded notation.  These functions all have the “same” formula:   for every a in A.  But they are technically different functions because they have different domains.

Inclusion function

If  (see inclusion), then there is an inclusion function  that takes every element in A to the same element.  In other words,  inc(a) = a for every element .  This fits the property of codomain that requires (in this case) that  because that is what “  ” means  every element of A is an element of B.

Usage

¨  The notation  for the identity function on A show which set A we are using, but the notation “inc” does not show either set.

¨  The notation “inc” is my own and is not common.  Other notations I have seen are:  and  

¨  Many mathematicians who use the looser definition of function never talk about the inclusion function.  For them, it is merely the identity function. 

Properties

The inclusion function is injective.  It is  surjective if and only if A = B.

Understanding the inclusion function

¨  The definition says the inclusion function “takes every element in A to the same element.”  I could have worded it this way: 

“The inclusion function  takes each element of A to the same element regarded as an element of B.”

This wording incorporates elements of “how you think about X” into the definition of X.   This is loose and unrigorous.  But I’ll bet a lot of readers would understand it more quickly that way! 

¨  The graph of inc is the same as the graph of  and they have the same domain, so that the only difference between them is what is considered the codomain (A for , B for the inclusion of A in B).   So inc is different from  if you require that functions with different codomains be different (discussed here).

Constant function

If A and B are nonempty sets and b is a specific element of B, then the constant function  is the function that takes every element of A to b; that is,  for all .

Usage

The notation  is not common.  There is no standard notation for constant functions.

Properties

The constant function is injective only if A has exactly one element.  It is  surjective only if B has exactly one element.

How to understand constant functions

¨  A constant function takes everything to the same thing.  It has a one-track mind. 

¨  A constant function from  to   has a horizontal line as its graph. 

¨  The constant function  is not the same thing as the element b of B. 

Empty function

If A is any set, there is exactly one function . Such a function is an empty function. Its graph is empty, and it has no values. An identity function does nothing. An empty function has nothing to do.

Properties

The empty function is vacuously injective.  It is  surjective only if A is empty.

Coordinate function

If A and B are sets, there are two coordinate functions (or projection functions)  and .  Thus for  and ,  and .

In general for an n-fold cartesian product, the function  takes an n-tuple to its i th coordinate. 

Properties