1The language of mathematics and prompting

1.1 The art of prompting

As of August 2024, there is a multitude of chatbots available on the internet. Some of them, like ChatGPT, Claude and Gemini (and Llama 3.1, Mistral, ... the list goes on) have quite impressive reasoning capabilities. These models are now multimodal i.e., they even accept non-textual input, such as images, sound and video. In principle you can upload a picture of a math exercise and the chatbot will provide a solution. Well, that is, on a good day and for a not too difficult exercise.The use of chatbots is encouraged throughout this course. In fact, they are even allowed during the exam. It is my hope that you will learn mathematics on a deeper level by communicating with the machine using carefully designed prompts - see the OpenAI guide on prompt engineering.In some places in the notes, you may see quite specific prompts, such as

Always answer in the context described below.
Context: I am taking a first semester college level course in introductory mathematics and optimization. Emphasis is on precision and small proofs in mathematics. Please give some conrete examples with mathematics involved. 

- Tell me what a good and precise prompt is.
- Show me two examples of good prompts.
- What is a good prompt if I want to know about code written in Sage or python?
- What is a good prompt if I want to know how to enter some specific mathematics in LaTeX?

A click on a chatbot link copies the prompt to the clipboard and takes you to the chatbot, where you can paste the prompt and get feedback.The prompt may also be hidden in a button as below. LLM There is a lot of wisdom baked into those weights! In this first chapter I will show a lot of prompts. In the following chapters less so. Here you are expected to prompt the chatbots on your own.Be careful! Prompting without any knowledge of the mathematics can be disastrous. I have seen exams written by generative AI spanning many pages with little or no relevant mathematical content. This is known as GIGO or garbage in garbage out. Good prompting comes from an understanding of the mathematics and often resembles concise and beautifully written instructions.

1.2 Black box warnings

Modern mathematics is perhaps not like anything you have encountered so far. It calls for a lot of focus and precision, especially when writing down solutions to problems. It is a bit like programming a computer. There is no room for imprecision and half-baked sentences.This course amounts to $10$ ECTS or approximately $280$ hours. Suppose that you spend a week studying for the exam, say $40$ hours. Lectures, exercise classes, and MatLab amount to $14\cdot (4 + 2 + 3)$ hours $= 126$ hours. This leaves around $114$ hours for your own study and immersion. Put in other terms, you are supposed to work around $8$ hours per week outside classes for this course. With classes, each week calls for $17$ hours of work. There is a very close relationship between the amount of hours you log each week and your result at the exam. To state the obvious: numbers don't lie. If you put in the time, you will almost certainly do well. Try to allocate time for IMO in your weekly schedule and please (ab)use all the help that is provided.

I am taking a first semester college level mathematics course spanning 14 weeks and 10 ECTS. One ECTS
amounts to 28 hours. The teaching activities every week amount to 4 hours of lectures,
2 hours of exercise sessions and 3 hours of study cafe. I expect that final exam will take 40 hours
of the 10 ECTS. Please schedule a weekly study plan for me along with a plan for the final exam.

1.3 Computer algebra (and python)

Computers are exceptionally fun, but be careful! Nothing really beats a clear thinking human mind. To wit, I asked WolframAlpha to solve a certain optimization problem and it came up with the answer 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 code One may also enter code in several other languages in the Sage input windows in the interactive notes. First adjust the prompt below according to your needs and get feedback from an LLM.

I am taking a mathematics course that uses the computer algebra system Sage.
The course uses the browser, where I can enter and run small snippets of code in Sage and python.
I have no/some/extensive prior programming experience.
Give me a brief introduction to Sage and explain how it relates to python.
Finish your reply with a small exercise I can do. 
If no/some/extensive is present above in this prompt, remark this and only reply with
Please select your programming experience.

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/. We need a precise setup for communicating mathematics. This involves the introduction of propositional logic, predicates and sets. Let me emphasize, that this is an introductory course and not a rigorous introduction to mathematics. As such it is an organic approach, where I hope that you will return and fill out the gaps instead of getting overwhelmed by formal details already from the beginning.However, the underlying goal is to show that mathematical precision and proofs are similar to constructing correct computer programs. In fact, this whole first chapter may be viewed as the beginning of a computer program, where we state the exact definitions for use in the following chapters. We begin by introducing sets and numbers.

1.4 Numbers

A set is a collection of (mathematical) objects or elements. When defining a set we use the symbols $\{$ and $\}$ to denote the beginning and end of its definition. For example, $\{\text{N, i, e, l, s}\}$ is the set of characters in my first name and $\{8, 0\}$ are the digits in the postal code for Aarhus C. The ordering in the listing of the elements is unimportant so that $$\begin{aligned} \{\text{N, i, e, l, s}\} &= \{\text{l, e, i, s, N}\}\\ \{8, 0\} &= \{0, 8\} \end{aligned}$$ are identical sets. If $S$ is a set, we will use the notation $x\in S$ to denote that $x$ is an element in $S$. For example, e$\in \{\text{N, i, e, l, s}\}$. Later, we will introduce much more detail about sets. For now, we just need the basic notation for defining them.Our fundamental mathematical objects in this course are numbers and we introduce them right away.

1.4.1 The natural numbers $\mathbb{N}$ and the integers $\mathbb{Z}$

The set of natural numbers is $$\mathbb{N} = \{1, 2, 3, \dots\}. \tag{1.1}$$ The set of integers is $$\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}. \tag{1.2}$$ These are infinite sets, since they contain infinitely many elements as indicated by the dots $\dots$It makes sense to add and multiply two integers $a$ and $b$. We will denote their addition or sum as $a + b$ and their multiplication or product as $a b$. A fundamental fact is that addition and multiplication are commutative i.e., $a + b = b + a$ and $a b = b a$.Please notice right away that expressions like $a + b + c$ and $a b c$ are complete nonsense for three integers $a, b$ and $c$. We only know how to add and multiply two integers, not three. A wonderful fact comes to our rescue: $$\begin{aligned} (a + b) + c &= a + (b + c)\\ (a b) c &= a ( b c). \end{aligned}\tag{1.3}$$ You get the same result no matter if you start adding (multiplying) $a$ and $b$ or $b$ and $c$ and then adding (multiplying) $c$ or $a$. So we may write $a + b + c$ and $a b c$ as a placeholder for one of the two ways of computing this expression in (1.3) .As you can see we use the symbol $+$ for addition, but no symbol for multiplication. This is the convention in (clean) mathematics as opposed to the $a * b$ coming from computer algebra (except perhaps in Mathematica or Wolfram language, where space is allowed for multiplication). However, when one of the factors is an actual number, we will use $\cdot$, so that for example $7$ times $9$ is written as $7\cdot 9$ and $a$ times $3$ is written $a\cdot 3$.

1.4.2 The rational numbers $\mathbb{Q}$

The natural numbers $\mathbb{N}$ and the integers $\mathbb{Z}$ are well defined by their representations in (1.1) and (1.2) . I will assume that you know how to add and multiply fractions, and that you do not make mistakes like $$\color{red} \frac{1}{2} + \frac{2}{3} = \frac{1+2}{2+3}=\frac{3}{5}.$$ In fact, if you temporarily forgot how to add fractions, you can use the wisdom in Definition 1.7 . You can replace $\frac{1}{2}$ by $\frac{3}{6}$ and $\frac{2}{3}$ by $\frac{4}{6}$ and then add the numerators as in $$\frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{3 + 4}{6} = \frac{7}{6}.$$ 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. In general, $$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d} \qquad\text{and}\qquad \frac{a}{b} \frac{c}{d} = \frac{a c}{b d}.$$

1.4.3 The real numbers $\mathbb{R}$

Informally (1.4) represents the real number $$d_0 + \frac{d_1}{10} + \frac{d_2}{100} + \cdots = d_0 + d_1 \cdot 10^{-1} + d_2 \cdot 10^{-2} + \cdots$$ given by an infinite set of digits as opposed to a rational number, which is given finitely by an integer (numerator) and a natural number (denominator). The format in (1.4) is the usual (floating point) output from your pocket calculator or computer algebra system. For example, $$\begin{aligned} \frac{1}{3} &= 0.333\dots\\ \frac{3}{7} &= 0.42857142857\dots\\ \frac{3}{14} &= 0.21428571428571428571\dots\\ \frac{22}{7} &= 3.14285714285714285714285714\dots\\ \frac{355}{113} &= 3.14159292035398230088495575\dots\\ \pi &= 3.14159265358979323846264338\dots \end{aligned}$$ The decimal expansion of $355/113$ above looks chaotic, but it eventually repeats itself after $112$ digits. In fact the decimal expansion of every rational number is periodic i.e., it repeats itself from a certain point.In the definition (1.4) of a real number, we are forced to make identifications as in the definition of a rational number. I will not go into details here, but just notice that the identification $$0.99999\dots = 1.000\dots$$ is forced upon us: if $$x = 9\cdot 10^{-1} + 9\cdot 10^{-2} + \cdots.$$ Then $$10 x = 9 + 9\cdot 10^{-1} + 9\cdot 10^{-2} + \cdots = 9 + x.$$ Therefore we must have $x = 1$.A real number that is not rational is called irrational. There are many more irrational numbers than rational ones. Famous ones are $\sqrt{2}, \pi$ and $e$. The irrational number $\sqrt{2}$ is a root in the polynomial $x^2-2$ with integer coefficients (it is an algebraic number). The numbers $e$ and $\pi$ are not even algebraic (they are transcendental).

1.4.4 Arithmetic rules for numbers

The long list of arithmetic rules above may seem complicated at first, but they are just a formal version of what you already know, such as for example, $\pi + 0 = \pi$, $2 + y = 0$ if $y = -2$ and $3 z = 1$ for $z = \frac{1}{3} = 3^{-1}$. Beware however, that the precision in Proposition 1.12 is necessary when programming a computer.The rules (ⅳ) and (ⅴ) are called the associative laws for addition and multiplication respectively. The rule (ⅵ) is called the distributive law. It connects multiplication with addition. We now move on to the (formal) language involved in reasoning about mathematics in general.

1.5 Propositional logic

A proposition is a (mathematical) statement that is true ($t$) or false ($f$). This could be a boolean expression in a computer program, like $1 < 2$.Sage Later we will see propositions with variables in them like $x < 2$. These are called predicates.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 at this point (see Example 1.31 ), but just point out that there are many explanations available online and, perhaps more importantly, refer you to Exercise 1.19 .

1.5.1 Propositional logic as a formal language

The entities $p$ and $q$ above may have real world interpretations like it rains or it is cloudy, but we will view them as variables that can be assigned the values true or false. Independent of this assignment we define a proposition as follows.

1.5.2 Truth tables and equivalent propositions

Given a proposition, it makes sense to substitute values ($t$ or $f$) for the variables and evaluate it using the rules in Definition 1.18 , since the parentheses leave no ambiguity as to how the evaluation must take place. For a given proposition in the variables $x_1, \dots, x_r$, there are $2^r$ ways of assignments to the set of variables. Each of these assignments results in the value true or false after evaluation. This is conveniently recorded in the truth table of the proposition as illustrated in the example below, where $r = 3$ so that the truth table has $2^3 = 8$ rows. The two propositions in Example 1.25 have identical truth tables. In general, if two propositions $P$ and $Q$ have identical truth tables we call them equivalent and write $$P \equiv Q.$$ In Example 1.25 we saw that $$(x_1 \land (x_2 \lor x_3)) \equiv ((x_1\land x_2) \lor (x_1\land x_3)).$$ The definition below is very important too keep in mind.

1.5.3 Using Sage to compute truth tables

Sage may be used to compute truth tables for propositions using the propositional calculus in Sage. Below is an example. Be sure to check how to enter $\land, \lor, \implies$ and $\neg$.

1.5.4 Variables, predicates and quantification

Variables are fundamental in computer programs. In the predicate $x = 1$, $x$ appears as a variable. Depending on what you substitute for $x$, the resulting proposition could be true or false or even meaningless. As an example, the latter case appears if $x$ is replaced by the character 'a'. This is what computer scientists call a type error. You cannot compare a character with a digit.

For every $\forall$ and there exists $\exists$

Suppose that we have a predicate $p(x)$, such that $p(x)$ is a proposition for $x\in S$, where $S$ is some set. Then we define the proposition $$\exists x\in S: p(x) \tag{1.6}$$ to be true if there exists $x\in S$, such that $p(x)$ is true. We let $$\forall x\in S: p(x) \tag{1.7}$$ be the proposition defined by $$\neg (\exists x\in S: \neg p(x)).$$ In other words, (1.7) says that $p(x)$ is true for every $x\in S$, since there does not exist $x\in S$ making $p(x)$ false. Le me be absolutely clear. To show that $\forall x\in S: p(x)$ is false, it is enough to find just a single $x\in S$ so that $p(x)$ is false. The following is an excerpt from the infamous Beredskabsprøve Datalogi.

1.5.5 Proofs and inference rules

A proof begins with an assumption $P$ and proceeds with a sequence of logical steps called inference rules leading to a conclusion $Q$.You have been taught how to solve equations in steps leading to a solution. Each step turns out to be an inference rule and the conclusion is the solution. Let us see how for the simple equation $x + 1 = 2$. Formally we want to prove $$\forall x\in \mathbb{R}: x + 1 = 2 \implies x = 1.$$ The first inference rule is $a = b \implies a + c = b + c$ i.e., we are allowed to add the same number to both sides of an equality. This implies $$x + 1 = 2 \implies (x+1)-1 = 2 - 1 = 1.$$ To be very precise we now use Proposition 1.12 (ⅳ) as an inference rule i.e., $$(x + 1) - 1 = 1 \implies x + (1 - 1) = 1.$$ Then we use Proposition 1.12 (ⅷ) to conclude $$x + (1 - 1) = 1 \implies x + 0 = 1$$ and then finally, we get by Proposition 1.12 (ⅶ) that $$x + 0 = 1 \implies x = 1.$$ So to solve the equation $x + 1 = 2$, we are actually using four (!) inference rules along the way.

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

As you have seen, $\implies$ and $\iff$ are applied to link propositions in a logical argument. For example, $$x + 1 = 2 \iff x = 1\qquad\text{or}\qquad \forall x\in \mathbb{Z}: x + 1 = 2 \iff x = 1.$$ However, for $x \geq 0 \implies x^2 \geq 0$ we cannot link the two propositions by $\iff$, simply because $\forall x\in \mathbb{Z}; x^2 \geq 0 \implies x \geq 0$ is false (for $x= -1$).

1.5.7 More on mathematical proofs

Most professional mathematicians rarely think about the precise definition of a proof and would probably feel uncomfortable defining a proof precisely. 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. These automated proof systems build on dependent type theory, which we will not go into.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 (1.14) , there is an imprecision here. Can you tell what it is? The beauty here is that we have verified for all odd natural numbers that their square is odd. Not just a finite selection like $3, 7, 11, 13$.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 in the court of mathematics (or science).

1.5.8 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 is key in the example below, where you are left with the feeling that you have been deprived of a fair and genuine proof. 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!

1.6 More on sets

Propositions are important, but are confined by the binary values of true and false. We would like to work mathematically with objects like integers, floating point numbers, neural networks, computer programs and so on.

1.6.1 Objects and equality

One of the cornerstones of modern mathematics is deciding when two objects are the same i.e., given two objects $A$ and $B$, deciding whether the proposition $A=B$ is true of false. Oftentimes an algorithm for evaluating $A=B$ is needed.You may laugh here, but this is not always that easy. Even though objects appear different they are the same as in, for example the propositions $$\frac{105}{189} = \frac{35}{63}\qquad\text{and}\qquad \sin\left(\frac{\pi}{2}\right) = 1.$$ The first proposition 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. The first proposition is true in a very precise way, since $105\cdot 63 = 189 \cdot 35$. For two objects $A$ and $B$ we will use the notation $A \neq B$ for the proposition $\neg (A = B)$.We have already defined a set (informally) as a collection of distinct objects or elements. We introduce some more set theory here. A set is also an object as described in section 1.6.1 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 again 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.

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., $$\emptyset = \{\}\qquad\text{and}\qquad |\emptyset| = 0.$$

1.6.2 The symbols $\in$ and $\notin$

The symbol $\in$ is ubiquitous in set theory (and mathematics). If $A$ is a set, then $$x\in A \tag{1.10}$$ is a proposition. It is true if $x$ is an element of or belongs to $A$. The notation $$x\notin A$$ is defined by the proposition $\neg (x\in A)$. Also, as a bit of short hand notation, we will write $$a_1, a_2, \dots, a_n\in A\qquad\text{for the proposition}\qquad (a_1\in A)\land (a_2\in A)\land \cdots \land (a_n\in A).$$ Belongs to ($\in$) is straightforward in python.

1.6.3 Subsets

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 an element of $B$. So $A\subseteq B$ is a placeholder for the proposition $$\forall x\in A : x\in B$$ 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 $B$.

1.6.4 Set-builder notation

If $S$ is a set and $p(x)$ a predicate for $x\in S$, then we build the subset $$\{x\in S \mid p(x)\}\subseteq S \tag{1.11}$$ of $x\in S$ such that $p(x)$ is true. Python 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 $\land$) In the exercises below we have sloppily used the ordering $<$ before it is formally defined. We apologize to the reader and refer to Section 1.7 .

1.6.5 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 $B$ i.e., $$A\cap B = \{x \mid (x\in A) \land (x\in B)\}.$$ This is illustrated in the socalled Venn diagram below. The union $A\cup B$ is the set consisting of the elements in $A$ or $B$ i.e., $$A\cup B = \{x \mid (x\in A) \lor (x\in B)\}.$$ This is illustrated in the Venn diagram below. Lastly, the difference $A\setminus B$ (between $A$ and $B$) consists of the elements in $A$ not contained in $B$ i.e., $$A\setminus B = \{x \mid (x\in A) \land (x\notin B)\}.$$ This is illustrated in the Venn diagram below. Python The following is an excerpt from the infamous Beredskabsprøve Datalogi.

1.6.6 Pairs, triples and tuples

Oftentimes we want to consider more than one variable as input to a predicate. It is convenient to group the variables into one object consisting of the variables. This is done using tuples.Given two sets $A$ and $B$, we combine two elements $a\in A$ and $b\in B$ into a pair $(a, b)$, which is an element of the Cartesian product $$A \times B = \{(a, b) \mid a\in A, b\in B\}.$$ of $A$ and $B$. This is a new set built from $A$ and $B$. Python 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 $$(a_1, a_2, \dots, a_n)\in A_1\times A_2\times \cdots \times A_n \tag{1.13}$$ 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.

1.7 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. The natural order $<$ on the natural numbers $\mathbb{N}$ is $$1 < 2 < 3 < \cdots$$ For the numbers $\mathbb{Q}$ and $\mathbb{R}$ it is less obvious how to define an order. Mathematical simplicity comes to the rescue here. It is enough to define what (the) positive numbers are! We want (the) positive numbers to satisfy the conditions below.

1.7.1 Ordering $\mathbb{Z}$

As we saw in Remark 1.70 , the natural order on $\mathbb{Z}$ is defined by $\mathbb{Z}_+ = \mathbb{N}$, so that $x < y$ if $y-x\in \mathbb{N}$ for $x, y\in \mathbb{Z}$. This completely agrees with our preconception that $$\cdots < -3 < -2 < -1 < 0 < 1 < 2 < \cdots \tag{1.14}$$ To be precise, writing $\cdots < -3 < -2 < -1 < 0 < 1 < 2 < \cdots$ is nonsense, since $<$ is only defined for two integers.

1.7.2 Ordering $\mathbb{Q}$

We define the positive rational numbers as $$\mathbb{Q}_+ = \left\{\frac{m}{n} \in \mathbb{Q} \middle| m > 0\right\} = \left\{1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \dots \right\}.$$ One can check that $\mathbb{Q}_+$ satisfies the conditions in Definition 1.69 . So formally we get Using Definition 1.74 , 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!

1.7.3 Ordering $\mathbb{R}$

For the real numbers we define the positive numbers as $$\mathbb{R}_+ = \{x\in \mathbb{R} \mid x = d_0.d_1 d_2 \dots, d_0 \geq 0, x \neq 0\}.$$ Even though we have not precisely defined addition and multiplication of the real numbers, we claim that this definition of $\mathbb{R}_+$ satisfies the conditions of Definition 1.69 .One may prove that for every $x, y\in \mathbb{R}_+$, there exists $N\in \mathbb{N}$, such that $N x > y$. This is the archimedean property of the real numbers. One other crucial property is the completeness of $\mathbb{R}$. It says that a non-empty subset $S\subseteq \mathbb{R}$ with an upper bound $B$ i.e., $\forall x\in S: x\leq B$, always has a smallest upper bound. The rational numbers do not share this property, since for example $$S = \{x\in \mathbb{Q} \mid x^2 < 2\}$$ does not have a smallest upper bound inside $\mathbb{Q}$.

1.8 Proof by induction

A precocious Gauss See the article Gauss's Day of Reckoning for some history of this anecdote. proved the formula $$1 + 2 + \cdots + n = \frac{n(n+1)}{2} \tag{1.15}$$ 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 (1.15) 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 $$\forall n\in \mathbb{N}: p(n).$$ An induction proof is a way of proving this statement by showing two things:

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

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. In general if $S$ is a subset of set with an order $\leq$, then $s\in S$ is called a first element if $$\forall x\in S: s \leq x.$$ A crucial rule (or axiom) is that every non-empty subset of $\mathbb{N}$ has a first element! Notice that this is false for $\mathbb{Z}$. Let us see how an induction proof plays out in the above example with the statement $p(n)$ that $$1 + 2 + \cdots + n = \frac{n(n+1)}{2}. \tag{1.16}$$ 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 (1.16) is true. Then we may add $n+1$ to both sides of (1.16) 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. The 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.

1.9 The concept of a function

A function is a crucial concept in mathematics. In Sage (actually python here) a simple function can be programmed like The 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. 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 ... The precise mathematical definition of a function in terms of sets is the following. 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.

1.9.1 When are two functions the same?

Suppose that $f(x) = x + 1$. This way of defining a function is a bit sploppy. The domain and codomain is not defined. The correct way of defining a function also includes defining its domain and codomain as in Definition 1.90 . Two functions $f, g$ are the same when they have the same domain $S$ and codomain $T$ and $f(x) = g(x)$ for every $x\in S$.

1.9.2 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: $$f(x) = \begin{cases} 0 &\text{if } x\leq 0\\ x^2 &\text{if } x > 0 \end{cases} \tag{1.19}$$ 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 as

1.9.3 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.

1.9.4 Functions from and into products

1.9.5 Injective and surjective functions

We now define three very important notions related to functions.

1.9.6 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 $$f^{-1}(y) = x. \tag{1.20}$$ 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 (1.20) . Therefore (1.20) really is a good and sound definition.

1.9.7 The preimage

Definition 1.110 is short and sweet. Here is a first example of the preimage.

1.9.8 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 $$f: A\rightarrow B, \tag{1.21}$$ 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 neurons To 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 $$p(x_1, \dots, x_n) = \begin{cases} 1 &\text{if } w_1 x_1 + \cdots + w_n x_n > b\\ 0 &\text{if } w_1 x_1 + \cdots + w_n x_n \leq b \end{cases} \tag{1.23}$$ 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 (1.23) led to better training of deep neural networks.