Інтуїція про функції як відображення

A function can be considered a map from one set to another. Here are a few examples:

$\bullet$ Let $f(x) = 2x$. Then $f$ is a function that maps the set of real numbers to the set of real numbers.

$\bullet$ Let $f(x) = 2x$, but where we restrict the domain of $f$ to integers. Then $f$ is a function that maps the set of integers to the set of even integers. It would also be correct to say that $f$ maps the set of integers to the set of integers; not every element in the second set needs to be mapped to.

$\bullet$ For each living person, consider that person's favorite positive integer. If a person doesn't have a favorite, then associate that person with the number 0. This gives us a function that maps the set of living people to the set of non-negative integers.

In fact, any function is a map from one set $A$ to another set $B$. The set $A$ is called the domain, and the set $B$ is called the codomain. The range is the set of elements of $B$ that get mapped to.

So, anytime we have a function, sets are in view. Therefore, it's useful to understand what sets are and how we handle them.

A simple way to imagine a function is a "black box" that spits something out whenever you put something in.

There's an important caveat, however: if you put the same input into the function, it always has to produce the same output.

Simple mathematical examples of functions are "add 1", or "multiply by 2", or "subtract 5 and cube".

Simple nonmathematical examples of functions are "take the first letter" or "find the capital of a country (according to the world's state in 2014)".

On the other hand, "find a country that starts with a letter" is NOT a function, because if you put in the letter "B", for example, there are more than a dozen countries that might come out.

Most of the functions we'll be studying in calculus will be written out as expressions of variables, but remember that functions explained using words or that don't involve numbers are still perfectly legitimate functions.

Where do sets come into the picture?

Well, without more restrictions on what you can put into the function, nonsensical things can happen. Take "multiply by 2": we can multiply $3$ by $2$ and always gets $6$, so it seems like a function, but how do you multiply a banana by 2?

So, we usually specify two sets with a function. These are the domain and codomain of the function, which are the sets of what you can possibly put into the function and what could possibly come out.

With mathematicians' favorite notation, it is denoted as follows: $f : A \to B$ means $f$ is a function that takes something from the set $A$ as an input and outputs something in the set $B$.

Now, we must require that $f$ can actually output something in $B$ for every element in $A$.

What are sets? Well.. they are defined as

A set is a collection of distinct objects called elements. Seems useless? It turns out to be that Mathematics itself is built on sets. Although it isn't completely clear how this could possible be true, it will make more sense by the end of this post.

Some Notable Sets

We have been working with one particular set since the time we started arithmetic, to be precise, the set $\mathbb{N}$ of natural number. We started off with four operations which were addition, subtraction, multiplication, and division. We say that $\mathbb{N}$ is closed under addition and multiplication because the sum or product of two elements in $\mathbb{N}$ is inside the set itself. However, this set is not closed under subtraction nor division. When we considered the subtraction problem, we resulted with the set $\mathbb{Z}$ of integers. And when we try to satisfy closure with division, we get the set $\mathbb{Q}$ of rational numbers.

The set $\mathbb{Q}$ of rational numbers seem as if it isn't missing anything, and even the Greeks thought so too. However, consider the following problem:

What is the solution to $x^2 = 2$ where $x\in\mathbb{Q}$ (this means that $x$ is in $\mathbb{Q}$.)

Well... that's interesting. Let's see if we can let $x = p/q$ and get anywhere. Substituting gives us $$\frac{p^2}{q^2} = 2\implies p^2 = 2q^2.$$ Hey! $p$ must be even. Let's substitute $p = 2p'$. $$4p'^2 = 2q^2\implies 2p'^2 = q^2.$$ Yes! We're getting somewhere. $q$ is also even... Wait... doesn't that mean $x$ can never be in simplest form?

In fact, we have just proven that there is no $x \in \mathbb{Q}$ that will satisfy the equation. So obviously, the Greeks were wrong, and now we need to somehow extend this set. Well, let's just add all the irrational numbers, and then we have the set $\mathbb{R}$ of real numbers. We can in fact extend this further. Considering the fact that $x^2 = -2$ has no solutions in $\mathbb{R}$, we extend this to the set $\mathbb{C}$ of complex numbers.

Maps

Now we know that there are these mathematical objects called sets, but we have no way of studying them. This is why a mapping, also known as a function, was created.

A mapping or a function $f$ that maps a set $A$ to another set $B$ (denoted as $f:A \to B$), takes every element from $A$ and maps it to some element in $B$. $A$ is called the domain of $f$ and $B$ is the codomain of $f$. The elements that are mapped onto by the function is called the range, which is the image of the domain).