Function: the Most Important Concept in Mathematics

A function needs the concept of a mathematical set. A set \(\mathbb{X}\) is a collection of well-distinguishable elements. You can think of a set as a bag full of all kinds of different toys (elements). In mathematics, it's usually the numbers that are in that bag. Exactly such bags with numbers are needed for the concept of a function. And these numbers are distinct, that is no equal numbers may occur in the set.

The elements of the set \(\mathbb{X}\) are written in curly brackets, separated by a comma:

The set \(\mathbb{X}\) consists of four elements. These do not have to be sorted in a particular way. In addition, as it MUST BE, no element (no number) occurs twice in the set.

For example, another set \(\mathbb{Y}\) might look like this:

There is also an empty set \(U\) which has no elements:

Or there can also be an infinite set \(N\) which has infinitely many elements:

You can then introduce, for example, a variable \(y\) which is a placeholder for one of the elements of the set \(\mathbb{X}\). Mathematically written down, \( y \in \mathbb{X}\) means that \(y\) stands for any element of \(\mathbb{X}\). It is said "\(y\) is an element of \(\mathbb{X}\)".

Let's look at the following set:

What does \( y \in \mathbb{X}\) mean? This means that the variable \(y\) can take the following values: \(y = 1\), \(y = 5\), \(y = 3\) or \(y = 6\). You may, of course, name the variable as you wish: \(y\) or \(x\) or even \(\psi\) (Greek letter "Psi"). It is just a placeholder for the elements of the set.

Let us now define the concept of a function \(f\). For this we need two sets \(\mathbb{X}\) and \(\mathbb{Y}\). What the sets actually contain for elements depends on the specific function. However, since we want to define a function in general, we do not specify the sets concretely. For the definition it is also unimportant what is in the sets. There are any numbers there, but which one does not matter.

A function \(f\) is defined by assigning to each element of one set \(\mathbb{X}\) some element of the other set \(\mathbb{Y}\). Thus, the two sets are not equal. We need to select a set where we need to assign ALL elements. For the other set, we can also leave some of its elements unassigned.

• We call the set \(\mathbb{X}\), where we have to assign all elements, the domain of a function.

• We call the other set \(\mathbb{Y}\), where not all elements have a partner in the other set, a codomain (or set of destination).

The function \(f\), its domain \(\mathbb{X}\), and its codomain \(\mathbb{Y}\) are notated as follows:

Now we have to concretely assign to each element \(x \in \mathbb{X}\) an element \(y \in \mathbb{Y}\). Then we defined a concrete function. The assignment is notated as follows:

That means: Take the element \(x\) from the domain \(\mathbb{X}\) and assign to this element the element \(y\) from the codomain. Which element \(y\) it is exactly depends on the specific function \(f\). This is indicated by the notation \(f(x)\). Here \(x\) is called the function argument and \(f(x)\) the function value.

Let us summarize: Domain \(\mathbb{X}\) and the codomain \( \mathbb{Y} \) together with the corresponding assignment rule \( f(x) = y\) define a function \(f: ~\mathbb{X} ~\rightarrow~ \mathbb{Y} \).

Since we do not need to assign value \(x\) (element of the domain \(\mathbb{X}\)) to each \(y\) value (element of the codomain \(\mathbb{Y}\)), some \(y\) elements remain unassigned. All elements \(y\) which have been assigned an element \(x\) form a set which we call image set \(\mathbb{im}(f)\). This set is a subset of \(\mathbb{Y}\): \(\mathbb{im}(f) \subseteq \mathbb{Y}\).

The image \(\mathbb{im}(f)\) of the function \(f\) is a subset of \(\mathbb{Y}\) containing all \(y\) elements from \(\mathbb{Y}\) which have been assigned an element \(x\).

The image set (that is, the set of all \(y\) elements to which an \(x\) element has been assigned) together with the associated \(x\) elements, forms a graph. The graph of a function \(f\) is also a set. Let us denote it by \(\mathbb{G}(f)\). However, in this set \(\mathbb{G}(f)\) there are not directly numbers \(x\), \(y\) in it, but tuples \((x,y)\) of numbers. With the tuple notation we indicate that the \(x\)- and \(y\)-elements, combined in the tuple, belong together. Mathematically, the graph set can be notated as follows:

Here \(\mathbb{X} \times \mathbb{Y}\) is a so-called cartesian product of two sets \(\mathbb{X}\) and \(\mathbb{Y}\). \(\mathbb{X} \times \mathbb{Y}\) is a set in which all tuples \((x,y)\) are in without having to satisfy the property \(y = f(x)\), as in the case of the graph:

The graph \(\mathbb{G}(f)\) of a function \(f\) can be visualized by plotting on one axis the \(x\) elements (we call it \(x\) axis). And on the other axis, which is perpendicular to the \(x\)-axis, all \(y\)-elements (we call it \(y\)-axis) are plotted, to which an \(x\)-element was assigned.

Consider the graph set from the above example (equation 15). If you now draw a horizontal line through a \(y\) point and a vertical line through the corresponding \(x\) point, this will create an intersection of the two lines. This intersection is marked. This is exactly what is done with all the other tuples of the graph set. In such a way you can visualize the function \(f\):

Three important properties of a function \(f\), from which further properties can be derived, are: Injection, Surjection, Bijection.

A function \(f\) is injective if each element \(x\) in \(\mathbb{X}\) is assigned a different \(y\) element in \(\mathbb{Y}\):

Mathematically, an injective function is defined as follows:

Translated it means: \(f(x_1)\) is a \(y\) element to which \(x_1\) in \(\mathbb{X}\) has been assigned. And \(f(x_2)\) is a \(y\) element to which \(x_2\) in \(\mathbb{X}\) was assigned. Now if the two \(y\) elements are equal: \(f(x_1) = f(x_2)\)then the corresponding \(x\) elements must also be equal: \(x_1 = x_2 \). If this is satisfied, then the function \(f\) is injective.

A function \(f\) is surjective if each element \(y\) in \(\mathbb{Y}\) has been assigned an \(x\) element in \(\mathbb{X}\). So the image set is equal to the codomain: \(\text{im}(f) = \mathbb{Y}\) if the function is surjective:

If a function \(f\) satisfies both the surjectivity property and the injectivity property, then the function is called bijective. So bijection is just a combination of surjection and injection under one term. Instead of saying: "The function \(f\) is injective and surjective" one says: "The function \(f\) is bijective". For physicists, bijective functions are the functions that cause the least problems!