Download Properties of Functions

abstractmath.org
help with abstract math
Produced by Charles Wells. Home
Website TOC
Website Index Blog
Back to Functions Head
Last edited 5/12/2017 7:27:00 PM
PROPERTIES OF FUNCTIONS
Injectivity
Definition: injective
A function F : A ® B is injective if for any elements a, a ' in the domain A,
if a ¹ a ' then F (a) ¹ F (a ') .
Useful rewordings of the definition:
 F : A ® B is injective if and only if for all a, a ' Î A , if F (a) = F (a ') then a = a ' .
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.
 F : A ® B is injective if and only if different inputs give different outputs.
 F : A ® B is injective if and only if its kernel is the diagonal.
 F : A ® B 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 word “unique”. 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 x a 2x is injective on the real numbers and in fact on the
complex numbers. Doubling different numbers gives different numbers.
 The squaring function x a x 2 is not injective, because for example 22 = (- 2)2 .
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
23 = (- 1- i 3)3 = (- 1+ i 3)3 = 8
 See Wikipedia for more examples and more discussion.
Understanding injectivity
No information loss
An injective function F : A ® B 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 ½ 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 F : Z ® Z be defined by F(n) = 2n , then
F(m + n) = F(m) + F(n) (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 F : R ® R 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 y = F( x) = x 3 (which is injective) with some horizontal
lines
1.5
1.0
0.5
1.0
0.5
0.5
1.0
0.5
1.0
1.5
b) On the other hand, G( x) = x 3 - x is not injective. Note that some horizontal
lines cut it more than once, but others cut it only once.
1.5
1.0
0.5
1.5
1.0
0.5
0.5
1.0
1.5
0.5
1.0
1.5
c) Horizontal lines don’t have to cut a function at all. This is part of y = x 2 .
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.
1.5
1.0
0.5
1.5
1.0
0.5
0.5
1.0
1.5
0.5
1.0
1.5
Surjectivity
Definition: surjective
A function F : A ® B is surjective if and only if
for every element b in the codomain B there is an element in the domain A for which
F (a) = b .
Useful rewordings of the definition:
 F : A ® B is surjective if and only if the image of F is the same as the codomain B.
 F : A ® B is surjective if and only if “every element of B comes from an element of
A.”
Examples

Let F : R ® R be the squaring function, so F ( x) = x 2 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 x 2 = - 2 , for example).

+
But if you define G : R ® R (where R + denotes the set of nonnegative reals) by
G( x) = x 2 , 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 F : A ® B 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 F : R ® R is surjective if every horizontal line crosses its graph (one or
more times). Check out the graphs (a) through (c) above: F ( x) = x 3 and
G(x) = x 3 - x are surjective onto R , but H ( x ) = x 2 is not surjective onto R . H is
surjective onto the set of nonnegative real numbers.