Functions

Definition of a Function

Given two sets a function is any subset of such that each pairs with exactly one . It is denoted as . It can also be described that a function is a mapping from any element in one set (the domain ) to exactly one element in another (the range ).

Codomain

The codomain is the set of all possible outputs as declared, not necessarily those actually produced. In notation , is the codomain. This is an important distinction especially in functional equations where it might specify that . Every integer must be a valid input and produce an but not every integer has to be obtainable for example is a valid solution despite ever integer except being obtainable.

Range

The range is simply the set of all possible outputs denoted as . The range is always a subset of the codomain. . In notation , is the range.

Image

The range is also called the image of the function . You can also say that the image of is .

Preimage

Let the preimage of an element is . Similarly, the preimage of a set is . (Note that the preimage is always a set). The preimage is defined for all values in the codomain for all functions but a function can have no preimage if it’s not surjective (the preimage is just ) or multiple preimages is it’s not injective.

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Functional Properties

Injective

A function is injective on domain if different inputs map to different outputs. Formally denoted as

It can also be state that no two elements in the domain share the same image.

Monic

A function is monic iff

Theorem A function then is monic is injective.

Proof: Forward direction Suppose is injective and . Where . Then we have . By injectivity we get since this is , we get

Reverse direction Fix a one element set . Then define where . Then . Hence is injective.

Surjective

is surjective if it covers the whole domain. Formally denoted as

A function is always surjective if .

Epic

A function is epic iff

Theorem A function then is epic is surjective.

Proof: Forward Direction Suppose is surjective and . Where . Then by surjectivity, . So we can find a suitable . Hence .

Reverse Direction - Proof by contrapositive Suppose is not surjective. Then . Pick such a and call it . Then fix a two element set . Then define

But as so . But since , is not epic.

Bijective

is bijective if it is both injective and surjective

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Parity

We call a function even and we call odd . If neither holds than has no parity. This terminology comes from the function where if is even and if is odd . This concept still holds for many functions as if is equal to some power series then the power series only contains even powers of likewise for odd functions. Proof

1. Let and assume for contradiction that is even. In it’s radius of convergence is absolutely convergent.
2. Then split into it’s even and odd sums.
3. Since is even then .
4. But as every power is even. Similarly
5. This leads us with hence contains no odd powers
6. Similar argument can be made for odd functions.

Constructing Even and Odd Functions

Given a function which has no parity, we can create two special functions which are even and odd functions respectively with the property that . We can think of this as taking the odd and even “components” of . And if is a power series are precisely the part of the power series with odd and even powers respectively. Making these functions is very easy.

It is trivial to check these functions are odd and even and that they have the property .

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Indicator Function

Let , the indicator function of denoted by is a function with domain and range defined as

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