1The language of mathematics

The introduction of $\texttt{ChatGPT}$ on November 30, 2022 took the world by storm. The use of generative AI or large language models is encouraged throughout this course. It is my hope that you will learn mathematics on a deeper level by communicating with the machine using careful prompt engineering. Good prompts seem to resemble clearly written text.First we need to introduce some formal notation to be able to talk clearly about mathematics and also in a reasonable way formulate mathematical models of the real world. In modern mathematics the concept of a set is crucial. It is a bit tricky to define this precisely. We will go on and define the basic concepts of what is called naive set theory.

1.2 Computer algebra

We will use the computer algebra system Sage in exploring and experimenting with mathematics. This means that you will have to write small commands and code snippets. Sage is built on top of the very wide spread language python and you can in fact enter Python codeOne may also enter code in several other languages in the Sage input windows in this text. Below is an example of a basic graphics command in Sage. Push the Compute button to evaluate.You can install Sage on your own computer following the instructions on https://www.sagemath.org/.

1.3 Objects or elements and the symbols $=$ and $\neq$

Mathematics can be broadly viewed as handling objects precisely according to a specific system of rules. The first element of precision is in distinguishing the objects and deciding when they are the same. This calls for notation. If two objects $x$ and $y$ are the same, we write $x = y$. If they are different we write $x\neq y$.You may laugh here, but identifying objects is really one of the fundamental tasks of mathematics. It is not always that easy. Even though objects appear different they are the same as in, for example $$\frac{105}{189} = \frac{35}{63}\qquad\text{and}\qquad \sin\left(\frac{\pi}{2}\right) = 1.$$ The first example above is an identity of fractions (rational numbers). The second is an identity, which calls for knowledge of the sine function and real numbers. Each of these identities calls for some rather advanced mathematics.

1.4 Sets

A set is (informally) a collection of distinct objects or elements. A set is an object as described in section 1.3 and it makes sense to ask when two sets are equal.An example of a set could be the set $\{1,2,3\}$ of natural numbers between $0$ and $4$. Notice that we use the symbol "$\{$" to start the listing of elements in a set and the symbol "$\}$" to denote the end of the listing. Notice also that (by our definition of equality between sets), the order of the elements in the listing does not matter i.e., $$\{1, 2, 3\} = \{2, 3, 1\}.$$ We are also not allowing duplicates like for example in the listing $\{1, 2, 2, 3, 3, 3\}$ (such a thing is called a multiset).An example of a set not involving numbers could be the set of letters $$S=\{A, n, e, x, a, m, p, l, c, o, u, d, b, t, h, s, r, i\}$$ used in this sentence. The number of elements in a set $S$ is called the cardinality of the set. We will denote it by $|S|$.To convince someone beyond a doubt (we will talk about this formally later in this chapter) that two sets $A$ and $B$ are equal, one needs to argue that if $x$ is an element of $A$, then $x$ is an element of $B$ and the other way round, if $y$ is an element of $B$, then $y$ is an element of $A$. If this is true, then $A$ and $B$ must contain the same elements.Sets may be explored using (only) python. This is illustrated in the snippet below.

1.4.1 The empty set

There is a unique set containing no or zero elements. This set is called the empty set and is denoted $\emptyset$ i.e., $$\emptyset = \{\}\qquad\text{and}\qquad |\emptyset| = 0.$$ Not surprisingly the empty set is reflected as the empty list in Sage. The empty list has zero elements.

1.4.2 Sets of numbers

A set could also be the natural numbers (yes, I want $0$ as a natural number: $0$ is very natural, although it came late historically) $$\mathbb{N} = \{0, 1, 2, 3, \dots\},$$ or the set of integers $$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}.$$ These sets are called infinite, since they contain infinitely many elements. Even though the natural numbers seem as easy as one, two three, they contain wonderful and deep mathematical mysteries, such as the nature and distribution of the prime numbers $2, 3, 5, 7, 11, 13, 17, \dots$. Also please respect, that the negative numbers like $-3, -1\in \mathbb{Z}$ have caused confusion for centuries.We also have the set $\mathbb{Q}$ of rational numbers (fractions) and the set $\mathbb{R}$ of real numbers. The real numbers contains all the possible numbers that we encounter in this course.We will not define the arithmetic operations (like addition and multiplication) on $\mathbb{Z}, \mathbb{Q}$ and $\mathbb{R}$ formally. I will assume that you know how to add and multiply fractions, and that you do not make mistakes like $$\color{red} \frac{1}{2} + \frac{2}{3} = \frac{1+2}{2+3}=\frac{3}{5}.$$ Similarly, I will assume that you know that a rational number stays the same, when the numerator and denominator is multiplied by the same non-zero integer. For example, $$\frac{1}{2} = \frac{3}{6}\qquad\text{and}\qquad \frac{2}{3} = \frac{4}{6}.$$ In fact, $$\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{3 + 4}{6} = \frac{7}{6}.$$ The computation above says that it is straightforward to add pizza slices of the same size (one sixth), but that you need to think a bit when adding one half pizza slice and two pizza slices of size one third.

1.4.3 Notation and rules for arithmetic operations

Please do not use the symbol for multiplication coming from your favorite computer algebra system. Nothing is worse than looking at notation like $$a*b + c*(b+d)\qquad \text{and}\qquad 12*14 \tag{1.1}$$ in a written assignment. In the language of mathematics (1.1) is written $$a b + c (b+d)\qquad \text{and}\qquad 12\cdot 14. \tag{1.2}$$ When using variables multiplication is simply an elegant small space and $\cdot$ is used with numbers. Also recall the important distributive law for handling expressions formally. It says that $$a (b + c) = a b + a c. \tag{1.3}$$ Notice that you can read the distributive law from right to left i.e., you may write $a (b + c)$ instead of $a b + a c$.

1.4.4 The symbols $\in$ and $\notin$

The symbol $\in$ is ubiquitous in set theory (and mathematics). It means belongs to or is an element of as in $x\in A$, where $x$ is an element and $A$ is a set. The symbol $\notin$ means is not an element of as in $x\notin A$ meaning $x$ is not an element of $A$.Belongs to ($\in$) is straightforward in python:

1.4.5 Subsets and the symbols $\subseteq$ and $\not\subseteq$

If $A$ and $B$ are sets, then $A\subseteq B$At times, the symbol $\subset$ is used instead of $\subseteq$. In our context these two symbols mean the same. However, the notation $A\subsetneq B$ means that $A\subseteq B$ and $A\neq B$. For example, $\{1, 2, 3\} \subseteq \{1, 2, 3\}$ and $\{1, 2, 3\} \subset \{1, 2, 3\}$. means that every element of $A$ is also an element of $B$. In this case we say that $A$ is a subset of $B$. We also use the notation $A\subsetneq B$ to indicate that $A\subseteq B$ and $A\neq B$. In this case we say that $A$ is a strict subset of $B$. MentimeterWe have for example that $$\mathbb{N} \subseteq \mathbb{Z}.$$ What does $A\not\subseteq B$ mean? Here we have to be a little careful. We want this notation to mean that $A$ is not a subset of $B$. In order for $A\subseteq B$ to be false, there must exist $x\in A$, such that $x\notin B$. This is the meaning of $A\not\subseteq B$. For example, $$\mathbb{Z}\not\subseteq \mathbb{N},$$ since $-1\in \mathbb{Z}$ and $-1\notin \mathbb{N}$.Below Sage (not python) will list all subsets of the set $\{1, 2, 3\}$. Before pressing the Compute button, try to write them down on your own.It turns out that the empty set $\emptyset$ is a subset of any set.Does this make sense? We will explain this later talking about the logical relation $\implies$.

1.4.6 Intersections, unions and the symbols $\cap,\,\, \cup$ and $\setminus$

Suppose that we have two sets $A$ and $B$. Then the intersection $A\cap B$ is the set consisting of the elements in both $A$ and $B$. This is illustrated in the socalled Venn diagram below.The union $A\cup B$ is the set consisting of the elements in $A$ or $B$. To be more precise, an element is in $A\cup B$ if it is in $A$ or in $B$ (or in both of them):Lastly, the difference $A\setminus B$ (between $A$ and $B$) consists of the elements in $A$ not contained in $B$:You should experiment using the python window below to get a feeling for these three operations.The following is an excerpt from the infamous Beredskabsprøve Datalogi.Mentimeter

1.4.7 Pairs, triples and tuples

Given two sets $A$ and $B$ we can form the new set $A\times B$, which is the set of pairs $(a, b)$, where $a\in A$ and $b\in B$. For example, $$\{1, 2\}\times \{1, 2, 3\} = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)\}.$$ The set $A\times B$ is also called the Cartesian product of $A$ and $B$.MentimeterThe Cartesian product can be computed in python as shown below.There is no need to restrict ourselves to pairs. We might as well consider triples $A\times B\times C$ i.e., the set of all $(a, b, c)$, where $A$, $B$ and $C$ are sets, or for that matter general tuples $$(a_1, a_2, \dots, a_n)\in A_1\times A_2\times \cdots \times A_n$$ of any length $n\in \mathbb{N}$, where $a_1\in A_1, a_2\in A_2, \dots, a_n\in A_n$. Based on the above example with tuples we have, $$\begin{aligned} &\{0\}\times\{1, 2\}\times \{1, 2, 3\} = \\ &\{(0, 1, 1), (0, 1, 2), (0, 1, 3), (0, 2, 1), (0, 2, 2), (0, 2, 3)\}. \end{aligned}$$ You may check this using the python snippet below.

1.5 Ordering numbers

Let us be a little rigorous and introduce the (usual) ordering on our numbers with addition and multiplication using almost full blown mathematical formalities. First the formal definition for two integers $x, y\in \mathbb{Z}$:Notice that $x = y$ implies that $x\leq y$ (and $y\leq x$). Along this line we also define $x < y$ if $x \leq y$ and $x\neq y$.You can see that this definition agrees with our preconception that $$\cdots < -3 < -2 < -1 < 0 < 1 < 2 < \cdots \tag{1.5}$$ To be precise, writing $\cdots < -3 < -2 < -1 < 0 < 1 < 2 < \cdots$ is nonsense, since $\leq$ is only defined for two integers in (1.4). Notice that the set of integers has huge holes. Given two integers $a, b\in \mathbb{Z}$, such that $a < b$, we cannot always find an integer $c\in \mathbb{Z}$ in between $a$ and $b$: $$a < c < b.$$ The set of rational numbers has the property that they do not have holes. We can always find an in between number such as $c$ above. But we need a precise way of comparing rational numbers. A way to explain precisely why for example $$\frac{2}{3}\,\, \leq \,\, \frac{5}{7}.$$ Of course, you can enter the two numbers on a computer and see that $\frac{2}{3}$ is approximately $0.67$ and $\frac{5}{7}$ is approximately $0.71$, but we aim for the mathematical precise definition.A rational number $\frac{a}{b}$ is represented by a numerator $a\in \mathbb{Z}$ and a denominator $b\in \mathbb{Z}$ with $b > 0$. We already know the criterion for two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ to be equalTechnically speaking we are defining a socalled equivalence relation identifying the infinitely many ways of writing a rational number into one.:We wish to compare the two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$ deciding precisely how they are ordered:MentimeterAs for the integers, we also define $x < y$ if $x \leq y$ and $x\neq y$ for two rational numbers $x$ and $y$.Using this definition, you can check that $\frac{2}{3} < \frac{5}{7}$, since $$2\cdot 7 < 3 \cdot 5.$$ An easy, but surprising, way of finding a rational number strictly between these two is adding their numerators and denominators: $$\frac{2}{3} < \frac{2 + 5}{3 + 7} < \frac{5}{7}.$$ We wil try to explain the first inequality in mathematical general terms going through a rather formal proof consisting of five steps. These steps are given in the quiz below. Your task is to drag from the left and drop them to the right in an order, so that the proof makes sense. After that you are supposed, on your own, to write down a precise proof of the second inequality.The exercise below shows that our trick for finding rational numbers in between two given rational numbers can be made into a machine for generating all positive rational numbers!

1.5.1 Subsets of numbers and first elements

In a set equipped with an order, it is intuitively clear what a first element should be. For example, the natural numbers $\mathbb{N}$ has $0$ as its first element. On the other hand the set $\mathbb{Z}$ of integers does not have a first element (it is "infinite to the left").In fact every non-empty subset $S\subseteq \mathbb{N}$ has a first element. This follows from a rather special property of $\mathbb{N}$: there can be only finitely many natural numbers smaller than a given one. This is not true for $\mathbb{Z}$. Here there are infinitely many integers smaller than any integer.Mentimeter

1.6 Propositional logic

We have seen quite a few mathematical statements that ended up being true or false. Such statements are called propositions. Here are two examples of propositions usings sets (in python):Propositions can be combined into new (compound) propositions. Take for example the propositions $$\begin{aligned} &p: \text{it rains}\\ &q: \text{it is cloudy}. \end{aligned}$$ Then ($p$ and $q$) is a perfectly good new proposition reading it rains and it is cloudy. The same goes for (if $p$ then $q$), which reads if it rains then it is cloudy. The proposition (if $q$ then $p$) reads if it is cloudy then it rains. This proposition is (clearly) false. We need some notation to describe these compound propositions: $$\begin{array}{ll} p \land q\qquad\qquad & \qquad\qquad p \text{ and } q\\ \\ p \lor q\qquad\qquad & \qquad\qquad p \text{ or } q\\ \\ p\implies q\qquad\qquad & \qquad\qquad \text{if } p \text{ then } q\\ \\ \neg p\qquad\qquad & \qquad\qquad \text{not } p \end{array}$$ The compound propositions are either true($t$) or false ($f$) depending on $p$ and $q$. The dependencies are displayed in the truth tables below.The tables for the compound propositions $p\land q, p\lor q$ and also $\neg p$ are not too hard to grasp. The table for $p\implies q$ raises a few more questions. Why is $f\implies t$ true? I will not go into this, but just point out that there are many explanations available online and, perhaps more importantly, refer you to Exercise 1.40 and the remark below.We may also check what Sage outputs for the truth table of $p\implies q$ (or for that matter any formulaNotice that Sage uses -> for $\implies$, \& for $\land$, | for $\lor$ and ~ for $\neg$.. combining the logical operations above):MentimeterIn the exercise below you will see for example that $p\implies q$ is the same as $\neg q \implies \neg p$.The notation $p \iff q$ is used frequently. It means that both $p\implies q$ and $q\implies p$ are true i.e., $$(p\implies q) \land (q\implies p).$$ Mentimeter

1.6.1 The symbols $\exists, \forall$ and propositions with variables (predicates)

In mathematics one usually reasons with propositions with variables, such as $x > 0$. A proposition with variables is usually called a predicate.In order to evaluate a predicate with a variable $x$, one must first specify to which set $S$ the variable belongs. For example, the predicate $p(x)$ given by $$x^2 > 0,$$ does not make sense if $x$ is taken from the set of letters in the English alphabet (not unless you give an interpretation of $x^2$, $>$ and $0$ in this set). However, if $x\in \mathbb{Z}$, then $p(x)$ certainly makes sense. Whether $p(x)$ is true depends on $x$.For example, $p(0)$ is false, whereas $p(-1)$ is true. This leads us to the existential and universal quantifiers $\exists$ and $\forall$. The former reads there exists and the latter for every.For example, the predicate $$\exists x\in \mathbb{Z}: \neg p(x)$$ is true and so is $$\forall x\in \mathbb{Z}\setminus\{0\}: p(x).$$ The symbol ":" above means "such that"Therefore $\exists x\in \mathbb{Z}: \neg p(x)$ reads "there exists $x$ in $\mathbb{Z}$, such that $\neg p(x)$ is true".. Also, $\forall x\in \mathbb{Z}: p(x)$ is false, because $\exists x\in \mathbb{Z}: \neg p(x)$ is true. Linking ":" with $\implies$ one may say that for a general predicate $q(x)$. This statement can only be false if there exists $x\in S$, such that $q(x)$ is false (see the truth table for $\implies$ above). In particular, $\forall x\in \emptyset: x = 17$ is a true statement!In general, So we do not really need the quantifier $\exists$, when we have $\neg$ and $\forall$, but $\exists$ is convenient and used all the time. Notice that $\exists x\in \emptyset: x = 17$ is a false statement!The quantifiers are important to learn and apply when expressing mathematical ideas. So is the use of predicates in writing up subsets: if $S$ is a set, $x$ a variable taking values in $S$ and $p(x)$ a predicate (making sense in $S$), then For example, if $q(x) = x^2 > 0$, then $$\{x\in \mathbb{Z} \mid q(x)\} = \mathbb{Z}\setminus \{0\}.$$ Suppose that $p_1(x), \dots, p_n(x)$ are predicates with a variable $x$ taking values in $S$, then we often use the notation (using $,$ instead of $\land$) The following is yet another excerpt from the infamous Beredskabsprøve Datalogi.

1.6.2 The use of implication ($\implies$) and bi-implication ($\iff$)

Usually $\implies$ and $\iff$ are applied to link propositions in a logical argument. An example is $$x \leq y \iff x + z \leq y + z$$ for integers $x, y, z$. To be completely precise, I should here write $$\forall x, y, z\in \mathbb{Z}: x \leq y \iff x + z \leq y + z, \tag{1.10}$$ but one often writes $\forall$ with words as for example for integers $x, y, z$.To be one hundred percent precise (1.10) means formally, that $$\forall x\in \mathbb{Z}: ( \forall y\in \mathbb{Z}: (\forall z\in \mathbb{Z}: x \leq y\iff x + z \leq y + z))).$$ Here $x \leq y\implies x + z \leq y + z$ is true and similarly $x + z \leq y + z \implies x \leq y$ (by using the definition (see (1.4)) of $\leq$ in $\mathbb{Z}$). So the use of $\iff$ is valid. A somewhat simpler example is $$x + 1 = 2 \iff x = 1\qquad\text{or}\qquad \forall x\in \mathbb{Z}: x + 1 = 2 \iff x = 1.$$ However, for $x \geq 0 \implies x^2 \geq 0$ we cannot link the two propositions by $\iff$, simply because $x^2 \geq 0 \implies x \geq 0$ is false (for $x= -1$).

1.7 What is a mathematical proof?

Most professional mathematicians rarely think about the precise definition of a proof. During many years of training they have assimilated knowledge by experience. Therefore many proofs seem born out of witchcraft containing several magical devices.However, many proofs appearing in respected mathematical journals, submitted by respected mathematicians, have turned out to contain errors. Recent developments in automated proof systems like Coq and LEAN show promise in checking proofs like for example the famous four color theorem.Informally a proof of a proposition $q$, consists in arguing that an implication $p\implies q$ is true by first assuming $p$. Usually this is done not only through one implication $p\implies q$, but through a series of intermediate implications $$p\implies q_1 \implies q_2 \implies q_3 \implies \cdots \implies q_N,$$ where the last proposition $q_N$ is $q$. If $p$ is true, this will constitute a proof that $q_N = q$ is true. Just like in (1.5), there is an imprecision here. Can you tell what it is?In this section we will illustrate a simple mathematical proof of the proposition: $$\forall n\in \mathbb{N}: p(n)\implies p(n^2),$$ where $p(n) = (n\text{ is odd})$ i.e., the square of an odd natural number has to be odd. This seems true for a first selection of examples: $3^2=9, 5^2=25, \dots$.First we need to know what $p(n)$ means. What does it mean exactly for a number to be odd? This means that it is not divisible by $2$ or that there exists another natural number $a$, such that $n = 2 a + 1$. So $$p(n) = \exists a\in \mathbb{N}: n = 2 a + 1.$$ Therefore we need to show that $$\left(\exists a\in \mathbb{N}: n = 2 a + 1\right) \implies \left(\exists b\in \mathbb{N}: n^2 = 2 b + 1\right).$$ Notice that I had to change $a$ into $b$ in the second proposition above. The two variables are not the same: $a$ is associated with $n$ and $b$ is associated with $n^2$.Let us assume that $n = 2 a + 1$. Now we need to argue that $n^2 = 2 b + 1$ for some $b\in \mathbb{N}$. You stare at this for a while and notice that we should use the assumption $n=2 a + 1$ in computing $n^2$: $$n^2 = (2 a + 1)^2 = (2 a)^2 + 2 (2 a) + 1^2 = 4 a^2 + 4 a + 1 = 2(2 a^2 + 2 a) + 1.$$ Thus, using our assumption we may conclude that if $n = 2 a + 1$, then $$n^2 = 2 b + 1,$$ where $b=2a^2+ 2 a$. This completes the proof.The beauty here is that we have verified for all odd natural numbers that their square is odd. Not just a finite selection like $3, 7, 11, 13$.Below I have given a very detailed walk through of the proof above. It exemplifies how to write up the proof mixing words and mathematics. In many ways a proof is like a detailed argument in a court case, except that the rules of mathematics are universal. You need the absolute truth.

1.7.1 Proof by contradiction

A proposition $p$ is either true or false. This seemingly obvious statement goes by the name of the law of excluded middle and dates back to the writings of Aristotle.The law of excluded middle can be turned into a powerful proof technique called proof by contradiction.Suppose we wish to establish that $p$ is true. Then we turn things upside down by assuming that $p$ is false i.e., that $\neg p$ is true. If we then by logical deduction can show that $$\neg p \implies q,$$ for some proposition $q$, which is demonstrably false, then $\neg p$ cannot be true (since true $\implies$ false is false). Therefore $\neg p$ must be false and $p$ must be true by the law of the excluded middle. This technique is used all the time!Perhaps the two most famous proofs by contradiction in all of mathematics are due to Euclid. The first one is about the infinitude of the prime numbers. Here one assumes to begin with that there are finitely many prime numbers and follows this through to a contradiction. See Exercise 1.56.MentimeterThe second one (perhaps even attributable to Artistotle) is about the irrationality of $\sqrt{2}$. Here one assumes to begin with that $\sqrt{2}$ is rational and follows this through reaching a contradiction. See Exercise 1.57.

1.7.2 Proof by induction

A precocious GaussSee the article Gauss's Day of Reckoning for some history of this anecdote. proved the formula $$1 + 2 + \cdots + n = \frac{n(n+1)}{2} \tag{1.11}$$ at the age of seven displaying remarkable ingenuity for his age. Lesser mortals usually use induction to prove this formula. Gauss was asked along with his classmates to compute the sum of all natural numbers $1, 2, \dots, 100$. Using his formula he quickly came up with the correct answer $5050$. His classmates had to work for the entire lesson. Suppose that the formula in (1.11) is viewed as a proposition $p(n)$. To prove the formula we need to prove it for all natural numbers (you can easily see that $p(1)$ and $p(2)$ are true) i.e., we need to prove $$\forall n\in \mathbb{N}\setminus\{0\}: p(n).$$ An induction proof is a way of proving this statement by showing two things:

1. $p(1)$
2. $\forall n\in \mathbb{N}\setminus\{0\}: p(n)\implies p(n+1)$

These two statements ensure that $p(1) \implies p(2)$. Therefore $p(2)$ must be true, since we assumed $p(1)$ true from the beginning. Similarly $p(2)\implies p(3)$ ensures that $p(3)$ is true and so on. In fact we have proved $p(n)$ for every $n\in \mathbb{N}$ using this technique. One can prove this using proof by contradiction and that every non-empty subset of $\mathbb{N}$ has a first element (see subsection 1.5.1 and below). Let us see how an induction proof plays out in the above example with the statement $p(n)$ that $$1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \tag{1.12}$$ Clearly $p(1)$ is true. We need to prove $p(n)\implies p(n+1)$, so we assume that $p(n)$ holds i.e., that (1.12) is true. Then we may add $n+1$ to both sides of (1.12) to get $$1 + 2 + \cdots + n + (n+1) = \frac{n(n+1)}{2} + (n+1).$$ Here the right hand side can be rewritten as $$\frac{n(n+1) + 2(n+1)}{2} = \frac{(n+1)(n+2)}{2},$$ which is exactly what we want. This is the conjectured formula for the sum of the numbers $1, 2, \dots, n, n+1$. Therefore we have proved that $p(n)\implies p(n+1)$ and the induction proof is complete.MentimeterThe last exercise related to induction concerns the famous pigeonhole principle. The statement itself looks innocent, well almost ridiculous, but it is very powerful. Even the go-to website mathoverflow for research mathematicians has a quite nice thread about this.

1.8 The concept of a function

A function is a crucial concept in mathematics. In Sage (actually python here) a simple function can be programmed likeThe code above seems to take a number and returns the number plus one. This (f) is in fact a function taking as input a number and returning as output the number plus one. Notice that we do not even know which numbers we are talking about here. In mathematics we need to have a more precise notion of a function. Mathematically a function $f$ takes values from a set $S$ and returns values in a set $T$. In details, it is denoted $f: S\rightarrow T$ and the value associated with $s\in S$ is denoted $f(s)\in T$.The above python function could more formally be denoted as $f: \mathbb{Z}\rightarrow \mathbb{Z}$ with $f(n) = n+1$ if we are dealing with the integers, but we cannot tell from the code.Well, to be fair ...If you want the super precise mathematical definition of a function, I will give it here. A function $f: S\rightarrow T$ is a subset $f\subseteq S\times T$, such that $(s, t_1)\in f \land (s, t_2)\in f \implies t_1 = t_2$. In words it states that a function $f: S\rightarrow T$ is a subset $f$ of $S\times T$, containing pairs having only one second coordinate for every first coordinate.The everyday working definition of a function is more intuitive: a machine taking input from some set $S$ and giving output in some set $T$. The uniqueness of the output is encoded in the mathematical definition of a function.Mentimeter

1.8.1 Notations for defining a function

If $f: S\rightarrow T$ is a function and $S$ is a finite set, then you can define $f$ using a simple table. This is best illustrated using an example. Suppose that $S = \{1, 2, 3\}, T = \mathbb{R}$ and $$\begin{aligned} f(1) &= \sqrt{2}\\ f(2) &= \pi\\ f(3) &= -1. \end{aligned}$$ Then $f$ is expressed in table form as $$\def\arraystretch{1.5} \begin{array}{c|ccccccc} x & 1 & 2 & 3\\ \hline f(x) & \sqrt{2} & \pi & -1 \end{array}$$ Very often the bracket (or Tuborg in Danish) notation is used. It is similar to if-then-else statements in programming: $$f(x) = \begin{cases} 0 &\text{if } x\leq 0\\ x^2 &\text{if } x > 0 \end{cases} \tag{1.15}$$ defines the function $f:\mathbb{R} \rightarrow \mathbb{R}$ that outputs $0$ if the input $x\leq 0$ and $x^2$ if $x>0$. In python we may express this asWe now define three very important notions related to functions.

1.8.2 Composition of functions

Given two functions $f: S\rightarrow T$ and $g: U\rightarrow V$, where $V\subseteq S$, we define a new function $f\circ g: U \rightarrow T$ by $$(f\circ g)(u) = f(g(u)).$$ This notion calls for some reflection. We have a total of four sets in this definition: $U, V, S$ and $T$ and, not to forget, the condition that $V\subseteq S$. If this last condition was not satisfied it would be meaningless to apply the function $f$ to $g(u)$. I hope the diagram below helps the understanding.

1.8.3 The inverse function

If $f:S\rightarrow T$ is bijective, then we may define a function $g: T\rightarrow S$, so that $(f\circ g)(y) = y$ for every $y\in T$ and $(g\circ f)(x)$ for every $x\in S$. This function is denoted $f^{-1}$.How do we define $f^{-1}(y)$ for $y\in T$? Well, since $f$ is surjective, we may find $x\in S$ so that $y = f(x)$. Now, we simply define $$f^{-1}(y) = x. \tag{1.16}$$ We cannot have $x_1 \neq x_2$ in $S$ with $f(x_1) = f(x_2) = y$, since $f$ is injective. We only have one choice for $x$ in (1.16). Therefore (1.16) really is a good and sound definition.

1.8.4 Neural networks

Having defined functions and composition of functions, we can deflate the term (deep) neural network, which is often clouded in magic and mystery.A neural network is a special case of a function $$f: A\rightarrow B, \tag{1.17}$$ where $A\subseteq \mathbb{R}^m$ and $B\subseteq \mathbb{R}^n$. Neural networks are often compositions of many intermediate functions called (hidden) layers.In a neural network the functions $f_1, f_2, \dots, f_n$ are viewed as neuronsTo be precise, the functions should be viewed as synapses. Depending on their input they either fire or do not fire a signal. Classically this is modelled by the perceptron, which is a function $p:\mathbb{R}^n\rightarrow \mathbb{R}$ of the form $$p(x_1, \dots, x_n) = \begin{cases} 1 &\text{if } w_1 x_1 + \cdots + w_n x_n > b\\ 0 &\text{if } w_1 x_1 + \cdots + w_n x_n \leq b \end{cases} \tag{1.19}$$ for fixed numbers $w_1, \dots, w_n$ (called weights) and a number $b$ (called the threshold). If the weighted sum $w_1 x_1 + \cdots + w_n x_n$ is above the threshold, the neuron fires (returns the value $1$). If not it does not fire (returns the value $0$).The output of one neuron can be used as input for other neurons in a potentially extremely complicated network:The diagram above represents a neural network, which is a function $\mathbb{R}^8\rightarrow \mathbb{R}^4$. This function is actually a composition (represented by the hidden layers $1$, $2$, $3$ and the output layer): $$\mathbb{R}^8\rightarrow \mathbb{R}^9 \rightarrow \mathbb{R}^9 \rightarrow \mathbb{R}^9 \rightarrow \mathbb{R}^4.$$ All of the nodes above, except the ones in the input layer, represent perceptrons.Mathematically there is no reason to use special functions such as perceptrons in each node. One also uses a (smooth) version of the perceptron employing the sigmoid function. With the notation above, this function is given as $$\sigma(x_1, \dots, x_n) = \frac{1}{1 + e^{-(w_1 x_1 + \cdots + w_n x_n) - b}}.$$ However, around 2011 it was observed that the perceptron activation function (ReLU) as defined in (1.19) led to better training of deep neural networks. It is today, the most popular activation function.