[Math] A function that is surjective but not injective, and function that is injective but not surjective

functionsproof-writingreal-analysis

$(b)$ $f$ is surjective

$(c)$ $|A| = |B|$

Right now I'm having trouble coming up with examples that would not contradict what I proved. If the function is going from A to A, then the cardinality of the domain and codomain are the same, and if it is either surjective or injective, then wouldn't it have to also be injective or surjective, respectively?

Best Answer

$f(x) = x^3-3x$ is surjective but not injective

$g(x) = e^x$ is injective but not surjective.

And both are functions from $\mathbb R \to \mathbb R$

Motivation

When I go hiking, I want to be able to retrace my steps and get back to my starting point.

When I go to a store, I want to be able to return home.

When I swim out from the shore, I want to be able to get back to the shore.

Going somewhere and then coming back to where one started is important in life.

And it is important in mathematics.

And it is important in functional programming.

Domain, Codomain, and Inverse

If a function maps a set of elements (the domain) to a set of values (the codomain) then it is often useful to have another function that can take the elements in the codomain and send them back to their original domain values. The latter is called the inverse function.

In order for a function to have an inverse function, it must possess two important properties, which I explain now.

The Injective Property

Let the domain be the set of days of the week. In Haskell one can create the set using a data type definition such as this:

data Day = Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday

Let the codomain be the set of breakfasts. One can create this set using a data type definition such as this:

data Breakfast = Eggs | Cereal | Toast | Oatmeal | Pastry | Ham | Grits | Sausage

Now I create a function that maps each element of the domain to a value in the codomain.

Here is one such function. The first line is the function signature and the following lines is the function definition:

f  ::  Day  ->  Breakfast
f  Monday       =  Eggs
f  Tuesday      =  Cereal
f  Wednesday    =  Toast
f  Thursday     =  Oatmeal
f  Friday       =  Pastry
f  Saturday     =  Ham
f  Sunday       =  Grits

An important thing to observe about the function is that no two elements in the domain map to the same codomain value. This function is called an injective function.

[Definition] An injective function is one such that no two elements in the domain map to the same value in the codomain.

Contrast with the following function, where two elements from the domain -- Monday and Tuesday -- both map to the same codomain value -- Eggs.

g  ::  Day  ->  Breakfast
g  Monday       =  Eggs
g  Tuesday      =  Eggs
g  Wednesday    =  Toast
g  Thursday     =  Oatmeal
g  Friday       =  Pastry
g  Saturday     =  Ham
g Sunday        =  Grits

The function is not injective.

Can you see a problem with creating an inverse function for g :: Day -> Breakfast?

Specifically, what would an inverse function do with Eggs? Map it to Monday? Or map it to Tuesday? That is a problem.

[Important] If a function does not have the injective property then it cannot have an inverse function.

In other words, I can't find my way back home.

The Surjective Property

There is a second property that a function must possess in order for it to have an inverse function. I explain that next.

Did you notice in the codomain that there are 8 values:

data Breakfast = Eggs | Cereal | Toast | Oatmeal | Pastry | Ham | Grits | Sausage

So there are more values in the codomain than in the domain.

In function f :: Day -> Breakfast there is no domain element that mapped to the codomain value Sausage.

So what would an inverse function do with Sausage? Map it to Monday? Tuesday? What?

The function is not surjective.

[Definition] A surjective function is one such that for each element in the codomain there is at least one element in the domain that maps to it.

[Important] If a function does not have the surjective property, then it does not have an inverse function.

[Important] In order for a function to have an inverse function, it must be both injective and surjective.

Injective + Surjective = Bijective

One final piece of terminology: a function that is both injective and surjective is said to be bijective. So, in order for a function to have an inverse function, it must be bijective.

Recap

If you want to be able to come back home after your function has taken you somewhere, then design your function to possess the properties of injectivity and surjectivity.

[Math] Is this function bijective, surjective and injective

The floor, $\lfloor x\rfloor,$ of a real number $x$ is defined as the largest integer less than or equal to $x$. For example;

$\lfloor 3.1\rfloor = 3$,
$\lfloor 3\rfloor = 3$ and
$\lfloor \pi \rfloor = 3$.

These examples demonstrate that the function $f:\mathbb{Q}\to \mathbb{Z}$ given by $f(x) = \lfloor x \rfloor$ is not injective since, for example, $\lfloor \pi \rfloor = \lfloor 3 \rfloor = \lfloor 3.1 \rfloor = 3$. This immediately implies that $f$ cannot be bijective (since a bijection is a function which is both injective and surjective).

However the function is surjective because any integer $k\in \mathbb{Z}$ is the image of itself. That is $f(k) = \lfloor k\rfloor = k$ for all $k\in \mathbb{Z}$.