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Functions

  Introduction

You’ve already encountered functions throughout your education.

Here, however, we will study functions on discrete domains and
ranges. Moreover, we generalize functions to mappings. Thus,
there may not always be a “nice” way of writing functions like
above.

Definition

Function
 
Definition

A function f from a set A to a set B is an assignment of exactly
one element of B to each element of A. We write f(a) = b if b is
the unique element of B assigned by the function f to the element
a ∈ A. If f is a function from A to B, we write

f : A → B

This can be read as “f maps A to B”.

Note the subtlety:

  Each and every element in A has a single mapping.
  Each element in B may be mapped to by several elements in
A or not at all.

Definitions

Terminology
 
Definition

Let f : A → B and let f(a) = b. Then we use the following
terminology :

A is the domain of f, denoted dom(f).
  B is the codomain of f.
  b is the image of a.
a is the preimage of b .
  The range of f is the set of all images of elements of A,
denoted rng(f).
Definitions

Visualization

A function, f : A → B.

Definition I

More Definitions
 
Definition

Let f1 and f2 be functions from a set A to R. Then f1 + f2 and
f1f2 are also functions from A to R defined by

Example
Definition II

More Definitions
 
Let and then

Definition

Let f : A → B and let The image of S is the subset of B
that consists of all the images of the elements of S. We denote the
image of S by f(S), so that

Definition III

More Definitions

Note that here, an image is a set rather than an element.
 
Example

Let

Draw a diagram for f.

The image of S is

Definition
Definition IV

More Definitions
 
A function f whose domain and codomain are subsets of the set of
real numbers is called strictly increasing if f(x) < f(y) whenever
x < y and x and y are in the domain of f. A function f is called
strictly decreasing if f(x) > f(y) whenever x < y and x and y are
in the domain of f.
Injections, Surjections, Bijections I

Definitions
 
Definition

A function f is said to be one-to-one (or injective) if



for all x and y in the domain of f. A function is an injection if it is
one-to-one

Intuitively, an injection simply means that each element in A
uniquely maps to an element in b.

It may be useful to think of the contrapositive of this definition:

Injections, Surjections, Bijections II

Definitions
 
Definition

A function f : A → B is called onto (or surjective) if for every
element b ∈ B there is an element a ∈ A with f(a) = b. A
function is called a surjection if it is onto.

Again, intuitively, a surjection means that every element in the
codomain is mapped. This implies that the range is the same as
the codomain.

Injections, Surjections, Bijections III

Definitions
 
Definition

A function f is a one-to-one correspondence (or a bijection, if it is
both one-to-one and onto.

One-to-one correspondences are important because they endow a
function with an inverse. They also allow us to have a concept of
cardinality for infinite sets!

Let’s take a look at a few general examples to get the feel for
these definitions.

Function Examples

A Non-function

This is not a function: Both and map to more than one
element in B.

Function Examples

A Function; Neither One-To-One Nor Onto

This function not one-to-one since and both map to . It is
not onto either since   is not mapped to by any element in A.

Function Examples

One-To-One, Not Onto

This function is one-to-one since every maps to a unique
element in B. However, it is not onto since is not mapped to by
any element in A.

Function Examples

Onto, Not One-To-One

This function is onto since every element is mapped to by
some element in A. However, it is not one-to-one since is
mapped to more than one element in A.

Function Examples

A Bijection

This function is a bijection because it is both one-to-one and onto;
every element in A maps to a unique element in B and every
element in B is mapped by some element in A.

Exercises I

Exercise I
 
Example

Let f : Z → Z be defined by

f(x) = 2x - 3

What is the domain and range of f? Is it onto? One-to-one?

Clearly, dom(f) = Z. To see what the range is, note that

Exercises II

Exercise I

Therefore, the range is the set of all odd integers. Since the range
and codomain are different , (i.e. rng(f) ≠ Z) we can also
conclude that f is not onto.

However, f is one-to-one. To prove this, note that

follows from simple algebra .

Exercises

Exercise II
 
Example

Let f be as before,

f(x) = 2x - 3

but now define f : N → N. What is the domain and range of f? Is
it onto? One-to-one?

By changing the domain /codomain in this example, f is not even a
function anymore. Consider

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