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# Functions

Introduction 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 Prev Next