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Last edited 7/15/2009 7:50:00 PM

Properties of functions

Injectivity

Definition: injective

 

A function  is injective if for any elements  in the domain A,

if  then .

Useful rewordings of the definition:

¨   is injective if and only if for all , if  then . This is the contrapositive of the definition, and so is equivalent to it.  In many cases it is the best version of the definition to use in a proof.

¨   is injective if and only if different inputs give different outputs. 

¨   is injective if and only if its kernel is the diagonal.

¨   is injective if and only if “different elements of A go to different elements of B.”

Usage

¨  An injective function is called an injection or is said to be one to one.

¨  It is the whole function that is or is not injective. 

Warning  Do not try to reword this definition using the wordunique”.  It is too easy to get it mixed up with the definition of functional property.

Examples

¨  The articles about most of the example functions state whether they are injective or not.  Checking each of these functions to see why it is or is not injective is an excellent way to get a feel for the concept of injectivity.

¨  The doubling function  is injective on the real numbers and in fact on the complex numbers.  Doubling different numbers gives different numbers.

¨  The squaring function  is not injective, because for example .  Squaring different numbers might give you the same number.

¨  The cubing function on the reals is injective.  It is not injective on the complex numbers, because for example

 

                                           

¨  See Wikipedia for more examples and more discussion.

Understanding injectivity

No information loss

An injective function  loses no information.  If you have an output from the function, you know it came from exactly one input.

¨   If you got 8 when you cubed a real number, the real number had to be 2.  The cubing function is injective on the reals. But if you got 8 when you cubed a complex number, the complex number could be any of three numbers (see above); the cubing function is not injective on the complex numbers.

¨  If you got 4 when you squared something, the something could have been either 2 or 2:  You lost some information about the input, namely its sign in this case.  The squaring function is not injective.

¨    If you get  as an output from the sine blur function, it could have come from any one of an infinite number of inputs.  The sin blur function is ridiculously noninjective.

Embeds as a substructure

Some functions preserve some structure.  For example, multiplying integers by 2 preserves addition.  In other words, if you let  be defined by , then  (write it out, don’t just believe me!).  This makes it a group homomorphism.  Because multiplying by 2 is injective, this says that the substructure of the group of integers with addition as operation that consists of the even integers is a copy of the group of integers itself. 

 

Horizontal line crosses the graph only once at most

Let  be a real continuous function.  Then F is injective if no horizontal line cuts it twice.  This is a useful way of thinking about injective continuous functions, but it doesn’t work with arbitrary functions.

Examples

a)   This is a plot of part of  (which is injective) with some horizontal lines

b)   On the other hand,  is not injective.  Note that some horizontal lines cut it more than once, but others cut it only once. 

c)   Horizontal lines don’t have to cut a function at all.  This is part of .  Horizontal lines below zero don’t cut its graph because 0 is its minimum.  There are horizontal lines that cut it twice, so it is not injective.

Surjectivity

Definition: surjective

A function  is surjective if and only if

 

for every element b in the codomain B there is an element in the domain A for which .

Useful rewordings of the definition:

¨       is surjective if and only if the image of F is the same as the codomain B.

¨       is surjective if and only if “every element of B comes from an element of A.”

Examples

¨      Let  be the squaring function, so  for every real number x.   Then F is not surjective, since for any negative number b, there is no real number a such that F(a) = b.  (You can’t solve , for example). 

¨      But if you define  (where  denotes the set of nonnegative reals)  by , then G is surjective.  As you can see, whether a function is surjective or not depends on the codomain you specify for it.   Note that “G is surjective” says exactly that every nonnegative real number has a square root.

Usage

Another way of saying that  is surjective is to say, “F is surjective onto B”, or simply, “F is onto B”.  Saying it this way does not depend on whether you use the loose or strict definition for functions. 

Example

A function  is surjective if every horizontal line crosses its graph (one or more times).  Check out the graphs (a) through (c) above:   and  are surjective onto , but  is not surjective onto .  H is surjective onto the set of nonnegative real numbers.

Exercise

How do you prove that a function F:A→B is not surjective?

Exercise

Let α be a relation on A.

Exercise

Show that if α is reflexive, then the coordinate functions p1 α :α→A and p2 α :α→A are surjective.

Exercise

Show that the converse of (a) need not be true.

Exercise

hard[ label: cantorex]  Show that there for any set S, no function from S to PS is surjective. Do not assume S is finite.

Extended hint: If F:S→PS is a function, consider the subset

 

No argument that says anything like "the powerset of a set has more elements than the set" can possibly work for this problem, and therefore such arguments will not be given even part credit. The reason is that we have developed none of the theory of what it means to talk about the number of elements of an infinite set, and in any case this problem is a basic theorem of that theory.

Let's be more specific: One such invalid argument is that the function that takes x to {x} is an injective function from S to PS, and it clearly leaves out the empty set (and many others) so PS has "more elements" than S. This is an invalid argument. Consider the function from N to N that takes n to 42n. This is injective and leaves out lots of integers, so does N have more elements than itself?? (In any case you can come up with other functions from N to N that don't leave out elements.)

Bijectivity

Bijectivity

Definition:bijective

[ label: bijdef] A function which is both injective and surjective is bijective.

Remark

A bijection F:A→B matches up the elements of A and B - each element of A corresponds to exactly one element of B and each element of B corresponds to exactly one element of A.

Usage

A bijective function is called a bijection and is said to be a one to one correspondence.

Example

For any set A, \idA :A→A is bijective. Another example is the function F:{1,2,3}→{2,3,4} defined by F(1)=3, F(2)=2, F(3)=4.

Exercise

[ label: mypbij]  Show that the function G:N→Z defined by

$$G(n)=
\begin{cases}
-\frac{n}{2} & \text{n even}\\
\frac{n+1}{2} & \text{n odd}
\end{cases}
$$

is a bijection.

Exercise

Show how to construct bijections β as follows for any sets A, B and C.

Exercise

β:A×B→B×A.

Exercise

β:(A×B)×C→A×(B×C).

Exercise

β:{1}×A→A.

Answer: (a) β=<x,y>&Verbar;→<y,x>:A×B→B×A (or you can write β(x,y)=<y,x> or β=λ<x,y>.<y,x>).

(b)  β= << x,y>,z>&Verbar;→<x,<y,z>>.

(c)  β= p2 (the second projection).

Exercise

Let α be a relation from A to B.

Exercise

Prove that α is functional if and only if the first coordinate function p1 α is injective. (See Section  [projfromrel] .)

Exercise

Prove that α is the graph of a function from A to B if and only if the first coordinate function is bijective.

Exercise

Give an example of a function F:R→ R++ for which F is bijective. ( R++ is the set of positive real numbers.) Answer: One possible answer is x&Verbar;→ ex .

Exercise

hard[ label: isexlast]  Give an example of a function F:R→ R+ for which F is bijective. ( R+ is the set of nonnegative real numbers.) Answer: Here is one possible answer:

$$f(x)=\begin{cases} 0 & \text{if $x=0$}\\ e^{x-1} &
\text{if $x\in \N$ and $x\neq 0$}\\ e^x & \text{if $x\notin
\N$}\end{cases}$$

Note that the image of x&Verbar;→ ex itself does not include 0.

Exercise

hard Let F:A→B be a function. Prove that the restriction to Γ(F) of the first coordinate function from A×B is a bijection. Answer: A typical element of Γ(F) is <x,F(x)> for some element  x of  A. Suppose p1 (x,F(x))= p1 (x',F(x')). We must show that <x,F(x)>=<x',F(x')>. By Specification  [opspec] , page , this requires us to show that x=x' and F(x)=F(x'). By definition of coordinate function ( [coordfunc] , page ), p1 (x,F(x))=x for any x. It follows that x=x'. But then by the singlevalued property ( [singlevalued] ), page , F(x)=F(x').

To see that p1 is surjective, suppose xA. By GS. [svvv] , page , there is yB such that <x,y>Γ(F). (In fact, y=F(x).) Then p1 (x,y)=x, so p1 is surjective.

Exercise

hard Prove that a subset C of A×B is the graph of a function from A to B if and only if the restriction to C of the first coordinate function is a bijection.

Exercise

hard[ label: alphastar]  Let β: Rel (A,B)→\fsA(PB) be the function which takes a relation α to the function α* :A→PB defined by α* (a)={bB&Verbar;aαb} (see Definition  [reltopow] ). Show that β is a bijection. (This function is studied further in Problem  [betanat] , page , and in Problem  [betainvp] , page .)

Exercise

hard[ label: prdprop]  Let A, B and C be sets. In this exercise we define a particular function β from the set \fsAB×\fsAC to the set \fsA(B×C), so that β
as input a pair of functions <f,g>, with f:A→B and g:A→C, and outputs a function β(f,g) from A to B×C. Here is the definition of β: for all a
A,


Prove that β is a bijection.

Permutations

Definition:permutation

[ label: permdef] A permutation of a set A is a bijection β:A→A.

Example

The fact just noted that \idA is a bijection says that \idA is a (not very interesting) permutation of A for any set A.

Example

[ label: permex]  The function F:{1,2,3}→{1,2,3} that takes 1 to 2, 2 to 1 and 3 to 3 is a permutation of {1,2,3}.

Usage

[ label: permexex]  Many books define a permutation to be a list exhibiting a rearrangement of the set {1,2,…,n} for some n. If the ith entry in the list is ai that indicates that the permutation takes i to ai .

Example

The permutation of Example  [permex] would be given in the list notation as <2,1,3>.

Worked Exercise

List all the permutations of {1,2,3,4} that take 1 to 3 and 2 to 4. Answer: <3,4,1,2> and <3,4,2,1>,

Exercise

List all six permutations of {1,2,3}.