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    Handbook of Elementary Mathematics
    Handbook of Elementary Mathematics
    Handbook of Elementary Mathematics
    Ebook186 pages1 hour

    Handbook of Elementary Mathematics

    By Simone Malacrida

    ()

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    About this ebook

    This book lays the foundations of mathematics, starting with logic and elementary operations and moving on to topics such as trigonometry, complex numbers, matrix and vector notations, while addressing plane, solid and analytic geometry, as well as the rudiments of combinatorial and numerical calculus. 
    Such topics are necessary for the understanding of mathematical analysis and all modern developments, while providing a useful extension of knowledge for an initial mathematical description of the natural phenomena around us.

    LanguageEnglish
    PublisherSimone Malacrida
    Release dateDec 23, 2022
    ISBN9798215394359
    Handbook of Elementary Mathematics
    Author

    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.

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      Book preview

      Handbook of Elementary Mathematics - Simone Malacrida

      ELEMENTARY MATHEMATICAL LOGIC

      ––––––––

      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.

      ––––––––

      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 the mathematical field, proportionality can also be defined, indicated with . 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 in this way , the uniqueness of the element in this way and the expression for each element is transcribed in this way .

      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.

      ––––––––

      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.

      ––––––––

      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:

      Idempotence property:

      Neutral element existence property:

      Inverse element existence property:

      Commutative property:

      transitive property:

      Associative property:

      Distributive property:

      ––––––––

      The concepts of equality, congruence, similarity, proportionality and belonging possess all these properties just listed. Ordering symbols satisfy only the transitive and reflexive properties; that of idempotence is satisfied only by also including the ordering with equality, while the other properties are not well defined. 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 just as logical connectors do .

      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:

      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.

      ––––––––

      ––––––––

      Logical implication and co-implication have such truth tables:

      ––––––––

      ––––––––

      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 process of logical implication 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. The logical co-implication can also be expressed as:

      ––––––––

      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

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