# 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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