The Book of Mathematics: Volume 1

By Simone Malacrida

Most of mathematics is presented in this book, starting from the basic and elementary concepts to probing the more complex and advanced areas.
Mathematics is approached both from a theoretical point of view, expounding theorems and definitions of each particular type, and on a practical level, going on to solve more than 1,000 exercises.
The approach to mathematics is given by progressive knowledge, exposing the various chapters in a logical order so that the reader can build a continuous path in the study of that science.
The entire book is divided into three distinct sections: elementary mathematics, the advanced mathematics given by analysis and geometry, and finally the part concerning statistics, algebra and logic.
The writing stands as an all-inclusive work concerning mathematics, leaving out no aspect of the many facets it can take on.

• Language: English
• Publisher: Simone Malacrida
• Release date: Dec 28, 2022
• ISBN: 9798201001407

Simone Malacrida

Simone Malacrida (1977) Ha lavorato nel settore della ricerca (ottica e nanotecnologie) e, in seguito, in quello industriale-impiantistico, in particolare nel Power, nell'Oil&Gas e nelle infrastrutture. E' interessato a problematiche finanziarie ed energetiche. Ha pubblicato un primo ciclo di 21 libri principali (10 divulgativi e didattici e 11 romanzi) + 91 manuali didattici derivati. Un secondo ciclo, sempre di 21 libri, è in corso di elaborazione e sviluppo.

Book preview

FIRST PART: ELEMENTARY MATHEMATICS

1

ELEMENTARY MATHEMATICAL LOGIC

Introduction

Mathematical logic deals with the coding, in mathematical terms, of intuitive concepts related to human reasoning.

It is the starting point for any mathematical learning process and, therefore, it makes complete sense to expose the elementary rules of this logic at the beginning of the whole discourse.

We define an axiom as a statement assumed to be true because it is considered self-evident or because it is the starting point of a theory.

Logical axioms are satisfied by any logical structure and are divided into tautologies (true statements by definition devoid of new informative value) or axioms considered true regardless, unable to demonstrate their universal validity.

Non-logical axioms are never tautologies and are called postulates.

Both axioms and postulates are unprovable.

Generally, the axioms that found and start a theory are called principles.

A theorem, on the other hand, is a proposition which, starting from initial conditions (called hypotheses) reaches conclusions (called theses) through a logical procedure called demonstration.

Theorems are, therefore, provable by definition.

Other provable statements are the lemmas which usually precede and give the basis of a theorem and the corollaries which, instead, are consequent to the demonstration of a given theorem.

A conjecture, on the other hand, is a proposition believed to be true thanks to general considerations, intuition and common sense, but not yet demonstrated in the form of a theorem.

Symbology

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Mathematical logic causes symbols to intervene which will then return in all the individual fields of mathematics. These symbols are varied and belong to different categories.

The equality between two mathematical elements is indicated with the symbol of , if instead these elements are different from each other the symbol of inequality is given by .

In the geometric field it is also useful to introduce the concept of congruence, indicated in this way and of similarity .

In mathematics, proportionality can also be defined, denoted by .

In many cases mathematical and geometric concepts must be defined, the definition symbol is this .

Finally, the negation is given by a bar above the logical concept.

Then there are quantitative logical symbols which correspond to linguistic concepts. The existence of an element is indicated thus , the uniqueness of the element thus , while the phrase for each element is transcribed thus .

Other symbols refer to ordering logics, i.e. to the possibility of listing the individual elements according to quantitative criteria, introducing information far beyond the concept of inequality.

If one element is larger than another, it is indicated with the greater than symbol >, if it is smaller with that of less <.

Similarly, for sets the inclusion symbol applies to denote a smaller quantity .

These symbols can be combined with equality to generate extensions including the concepts of greater than or equal and less than or equal .

Obviously one can also have the negation of the inclusion given by .

Another category of logical symbols brings into play the concept of belonging.

If an element belongs to some other logical structure it is indicated with , if it does not belong with .

Some logical symbols transcribe what normally takes place in the logical processes of verbal construction.

The implication given by a hypothetical subordinate clause (the classic if...then) is coded like this , while the logical co-implication (if and only if) like this .

The linguistic construct such that is summarized in the use of the colon:

Finally, there are logical symbols that encode the expressions and/or (inclusive disjunction), and (logical conjunction), or (exclusive disjunction).

In the first two cases, a correspondent can be found in the union between several elements, indicated with , and in the intersection between several elements .

All these symbols are called logical connectors.

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Principles

There are four logical principles that are absolutely valid in the elementary logic scheme (but not in some advanced logic schemes).

These principles are tautologies and were already known in ancient Greek philosophy, being part of Aristotle's logical system.

1) Principle of identity: each element is equal to itself.

2) Principle of bivalence: a proposition is either true or false.

3) Principle of non-contradiction: if an element is true, its negation is false and vice versa. From this it necessarily follows that this proposition cannot be true

4) Principle of excluded middle: it is not possible that two contradictory propositions are both false. This property generalizes the previous one, since the non-contradiction property does not exclude that both propositions are false.

Property

Furthermore, for a generic logical operation the following properties can be defined in a generic logical structure G (it is not said that all these properties are valid for each operation and for each logical structure, it will depend from case to case).

reflective property :

For each element belonging to the logical structure, the logical operation performed on the same element refers internally to the logical structure.

Idempotence property :

For each element belonging to the logical structure, the logical operation performed on the same element results in the same element.

Neutral element existence property :

For each element belonging to the logical structure, there is another element such that the logical operation performed on it always returns the starting element.

Inverse element existence property :

For each element belonging to the logical structure, there is another element such that the logical operation performed on them always returns the neutral element.

Commutative property :

Given two elements belonging to the logical structure, the result of the logical operation performed on them does not change if the order of the elements is changed.

transitive property :

Given three elements belonging to the logical structure, the logical operation performed on the chain of elements depends only on the first and last.

Associative property :

Given three elements belonging to the logical structure, the result of the logical operation made of them does not change according to the order in which the operations are performed.

Distributive property :

Given three elements belonging to the logical structure, the logical operation performed on a group of two of them and on the other is equivalent to the logical operation performed on groups of two.

The concepts of equality, congruence, similarity, proportionality and belonging possess all these properties just listed.

Ordering symbols satisfy only the transitive and reflexive properties.

In this case, the idempotence property is satisfied only by also including the ordering with equality, while the other properties are not well defined.

The logical implication satisfies the reflexive, idempotence and transitive properties, while it does not satisfy the commutative, associative and distributive ones.

On the other hand co-implication satisfies all of them as do logical connectors such as logical conjunction and inclusive disjunction.

An operation in which the reflexive, commutative, and transitive properties hold simultaneously is called an equivalence relation .

In general, De Morgan's two dual theorems hold :

These theorems involve the definitions of logical connectors and the distributive property.

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Boolean logic

For logical connectors it is possible to define, with the formalism of the so-called Boolean logic, truth tables based on the true or false values attributable to the individual propositions.

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The negation is true if the proposition is false and vice versa.

The logical conjunction is true only when both propositions are true.

The inclusive disjunction is false only when both propositions are false.

Exclusive disjunction is false if both propositions are false (or true).

The logical implication is false only if the cause is true and the consequence is false.

Logical co-implication is true if both propositions are true (or false).

In case the logical implication is true, A is called a sufficient condition for B, while B is called a necessary condition for A.

The logical implication is the main way to prove theorems, considering that A represents the hypotheses, B the theses, while the logical implication procedure is the proof of the theorem.

Logical co-implication is an equivalence relation.

In this case A and B are logically equivalent concepts and are both necessary and sufficient conditions for each other.

Recalling the exposed properties, the logical co-implication can also be expressed as:

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Applications of logic: proof of theorems

The mathematical proof of a theorem can be based on two large logical categories.

On the one hand there is the deduction which, starting from hypotheses considered true (or already demonstrated previously), determines the validity of a thesis by virtue of the formal and logical coherence of the demonstrative reasoning alone. Generally , following this pattern, a mechanism is applied that reaches from the universal to the particular.

On the other hand, we have the induction which, starting from particular cases, abstracts a general law. As repeatedly highlighted throughout the history of logic, every induction is actually a conjecture and therefore, if we want to use the inductive logical method, these propositions are to be considered axioms.

In modern logic, which we will not go into in this paragraph as it deals with advanced concepts far beyond the scope of these simple elementary bases, the inductive method is not accepted as the correct logical reasoning to prove theses mathematically.

The deductive method is therefore the main method of mathematical proof.

It is distinguished in the direct method, in which the thesis is actually demonstrated starting from the hypotheses, and in the indirect method, in which the thesis is assumed to be true and the logical path is reconstructed backwards to reach the hypotheses.

The indirect method can, in turn, make use of the proof by contradiction which, by denying the thesis, leads to a logical contradiction and therefore the thesis remains proved for the principle of the excluded middle.

The method by contradiction therefore consists not in proving that it is true, but that it is false.

Sometimes, one can resort to the proof of the so-called contranominal to arrive at the proof of the theorem.

This originates from the following logical relationship.

If it is true , then it is necessarily true too .

In some particular sectors of mathematics, for example in geometry, particular demonstrative constructs such as those of similarity and equivalence can be used.

Logical demonstration procedures are constructive and iterative, in the sense that previous results can be used to demonstrate new theses (this is the case of lemmas and corollaries for example) or the same logical procedures can be used a sufficient number of times to reach the proof of the thesis.

Finally, we point out that mathematical theorems, precisely because they have to be proved, are neither true nor false in absolute terms; it is the hypotheses that determine the veracity or otherwise of the theses.

Precisely for this reason, a general extension of mathematical knowledge is given by the mechanism of the weakening of hypotheses.

Given a general thesis proved under suitable hypotheses, which of the latter can be relaxed to obtain the same thesis?

If, on the other hand, other hypotheses are changed, what new theses can be deduced?

These are the main questions that lead to overcoming previous knowledge in both logic and mathematics.

Applications of Boolean logic: electronic calculators

Boolean logic, also called Boolean algebra, is the basis of modern electronic calculators.

Indeed, a computer memory, or a processor of the same or of a smartphone, is based on single units that are connected through logical operations.

In electronic calculators, every single command is encoded by high-level languages (for example operating systems) which in turn are based on medium-level programming codes.

These codes are mediated by other programs which act directly on the physical part of the machine.

The heart of every electronic calculator is given by a logic unit capable of coding and executing a large number of logical operations per second.

In electronics, logical operations are defined as follows:

- the negation is called NOT

- the logical conjunction is called AND

- the inclusive disjunction is called OR

- the exclusive disjunction is called XOR.

Furthermore, the negations of the previous ones are called NAND, NOR and XNOR.

Electronic calculators are made up of billions of elementary logic cells, each of which encodes one of these logical operations.

The binary numbering system, which has only two digits 0 and 1, is very well suited to interpreting Boolean logic. The digit 0 corresponds to the status of false, to the digit 1 the status of true.

In computer science, these digits are called bits.

Physically, the false state is made up of an unbiased circuit (that is, without the application of an electric voltage), while the true state is constituted by a polarized circuit.

Thus applying a direct reference voltage (for many years it was 5 volts continuously, but today there is a tendency to decrease this value from 3.3 volts to 2.1 up to 1.8 or 1.3 or 0.9 volts ), it is possible to identify the different logical states and build the physical foundations of an electronic calculator.

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Insight: syllogism and mathematical logic

The syllogism develops around this reasoning divided into three statements:

First proposition: all men are mortal.

Second proposition: Socrates is a man.

Third statement: Socrates is mortal.

Translated with the symbology of mathematical logic it becomes (called A the set of all men, b the identifying element of Socrates and C the fact of being mortal):

It is clear that, logically, this reasoning is flawless.

The real problem lies precisely in the first statement.

To say all men are mortal is in itself already knowing that Socrates, as a man, is mortal. In other words, the first statement derives from an induction already known a priori and, as such, it is a conjecture that cannot be demonstrated, but taken as true (common sense tells us that this is the case).

As such, the syllogism, being a reasoning based on a first inductive statement, does not generate real knowledge.

At the end of the third sentence we know that Socrates is mortal, but in reality we already knew it at the beginning, since, in order to be able to affirm that all men are mortal, we had to necessarily have already included Socrates himself.

Modern logic disregards the use of the syllogism to enrich knowledge, relying on other logical constructs, based on the deduction and demonstration of theorems.

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Exercises

Exercise 1

Prove De Morgan's first theorem using logical properties.

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De Morgan's first theorem states that:

Applying the distributive property of negation with respect to logical conjunction we arrive at the result of De Morgan's theorem.

Similarly, the second theorem is proved.

An alternative method of proof is to use truth tables.

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Exercise 2

Construct the truth table for the following logical construct.

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By applying the inclusive disjunction to the two tables it is clear that the logical construct is always true.

It is therefore a tautology.

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Exercise 3

Justify, through Boolean logic, the veracity of the method of demonstrating the contranominal.

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The method of demonstrating the contranominal is based on denying the thesis and demonstrating that this denial implies the denial of the hypotheses.

In logical terms it means admitting that if it is true , then it is necessarily true too .

From Boolean logic we know that the logical implication is false only if the cause is true and the consequence is false.

As can be seen, the two truth tables coincide.

2

ELEMENTARY ARITHMETIC OPERATIONS

Introduction

In addition to logic, the mathematical alphabet relies on numbers which are conceptual abstractions to encode the different quantities of a given element.

Almost all numerical alphabets, such as ours, are based on characters given by numbers; in our decimal number system, the digits are ten, including zero which indicates zero quantity.

A number is given by the composition of several digits; starting from the right, the last digit represents the units, the penultimate the tens, the antepenultimate the hundreds, the fourth last the thousands.

We can define elementary operations related to each numerical alphabet, for ease of use we only consider the decimal system that we use extensively.

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Addition and subtraction

Addition takes into account the increase of one quantity by another (or others).

The individual quantities are called addends, while the result of the addition is called the sum.

For addition, the commutative and associative properties are valid, furthermore the neutral element is given by zero.

Addition also satisfies an ordering property since the sum is always greater than the single addends and, conversely, each addend is always less than the sum.

The mathematical symbol for addition is +.

Subtraction, on the other hand, takes into account the reduction of one quantity by another (or others).

The quantity to be subtracted is called the minuend, the quantity to be subtracted is called the subtrahend, while the result is called the difference.

For subtraction the associative property holds, the neutral element is always given by zero and an ordering property is satisfied being the difference always smaller than the minuend and, vice versa, the minuend always greater than the difference.

The mathematical symbol of subtraction is minus –.

A special case of subtraction occurs when the subtrahend is greater than the minuend.

In this case, the difference is negative, i.e. less than zero.

Negative numbers are exactly the same in shape as positive numbers except with the prefix - .

In doing so we see that the subtraction does not satisfy the commutative property, but another one called anti-commutative:

This formulation allows us to unify the concepts of addition and subtraction.

We can associate the + and – signs with individual numbers and not with the operation.

Therefore subtraction is an addition between a positive and a negative number, applying the well-known rule of signs according to which an even number of agreeing signs (two plus or two minus) returns a positive sign, while an even number of discordant signs (a plus and a minus) results in a negative sign.

It happens vice versa if there are odd numbers of agreeing and discordant signs.

In this unifying view, the commutative property is always valid since subtraction falls into addition. Furthermore, each number has an inverse with respect to the addition/subtraction operation given by its negative counterpart.

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Multiplication and division

Multiplication is an operation that summarizes the iterated addition of equal numbers.

The numbers to be multiplied are called factors, while the result is called product.

The multiplication symbol is given by , even if in mathematics the dot is more often used or the multiplication symbol is totally omitted (and this happens in the vast majority of cases).

For multiplication the commutative and associative properties hold, furthermore the distributive property holds with respect to addition and subtraction:

The neutral element is given by unity (every number multiplied by 1 always gives itself), while the rule of the signs previously exposed is always valid.

Furthermore, for multiplication there is also a zero element, given precisely by zero (every number multiplied by zero always gives zero).

Division is the reverse of multiplication.

The number to be divided is called the dividend, the number that divides is called the divisor and the result is called the quotient.

The division symbol is given by , sometimes the slash / is also used.

The properties listed for multiplication do not apply to division.

The neutral element is given by unity (every number divided by 1 always gives itself) and the rule of signs previously exposed is always valid.

However, the division by zero operation is not defined.

If the dividend is greater than the divisor, the quotient is greater than 1, and if the dividend is less than the divisor, the quotient is less than 1.

The division operation brings up non-integer numbers, i.e. numbers that can only be defined using digits smaller than the unit.

For such figures, the convention of placing a comma between the upper part and the lower part is used.

The digits after the decimal point respectively express tenths, hundredths, thousandths and so on.

When the dividend is a multiple of the divisor, the quotient is an integer and is called quota and, in this case, the dividend is divisible by the divisor, the remainder being zero.

A number is always divisible by itself (giving the value 1) and by 1 (giving the value itself).

Numbers that are divisible only by themselves and 1 are called prime numbers.

Numbers that are divisible by 2 are called even, those that are not divisible by 2 are called odd.

A quotient can always be expressed as the sum of a quota and a remainder.

Non-integer quotients can have a limited number of decimal digits or an infinite number of such digits.

In the latter case, we speak of periodic numbers as the decimal digits (all or part of them) are always repeated in the same sequence.

Periodicity is indicated with a sign above the periodic digit or digits.

For example, the quotient obtained from the division between 1 and 3 is given by a number having infinite decimal digits all equal to 3 and is indicated as follows .

Another way of expressing division is to use the concept of fraction.

In this case, the dividend and the divisor are called numerator and denominator, respectively.

Since division by zero is not defined, the denominator of a fraction can never be equal to zero.

A fraction is indicated with the symbol of fraction ––, the numerator goes in the upper part, the denominator in the lower part.

A fraction is said to be reduced to its lowest terms, or irreducible, if the numerator and denominator are prime numbers, i.e. they are no longer divisible by each other, resulting in a fraction.

If the numerator is greater than the denominator, the fraction is greater than 1 and is said to be improper.

Conversely, it is less than 1 and is called a proper one.

Finally, the fraction is apparent if the numerator is a multiple of the denominator (because, in this case, the fraction is actually an integer).

We define the reciprocal of a number as that number which, multiplied by the first, always gives 1.

In other words, the reciprocal of a number is its inverse element with respect to multiplication.

With this definition and with the notation of fractions, we can unify the