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As in algebra, in order to avoid the overuse of parentheses one canmake a convention about the order of operations. To that end, we shall agree on thefollowing order of priorities for the symbols:¬,∧,∨,⇒,⇔.With this convention the expression ¬A ∧ B ∨ C ⇒ D should be interpretedas (((¬A) ∧ B) ∨ C) ⇒ D, and the relation A ∨ B ⇒ C as (A ∨ B) ⇒ C, not asA ∨ (B ⇒ C).We shall often give a different verbal expression to the notation A ⇒ B, whichmeans that A implies B, or, what is the same, that B follows from A, saying thatB is a necessary criterion or necessary condition for A and A in turn is a sufficientcondition or sufficient criterion for B, so that the relation A ⇔ B can be read in anyof the following ways:A is necessary and sufficient for B;A holds when B holds, and only then;A if and only if B;A is equivalent to B.Thus the notation A ⇔ B means that A implies B and simultaneously B implies A.The use of the conjunction and in the expression A ∧ B requires no explanation.It should be pointed out, however, that in the expression A ∨ B the conjunctionor is not exclusive, that is, the statement A ∨ B is regarded as true if at least oneof the statements A and B is true.
For example, let x be a real number such thatx 2 − 3x + 2 = 0. Then we can write that the following relation holds: 2x − 3x + 2 = 0 ⇔ (x = 1) ∨ (x = 2).1.1.2 Remarks on ProofsA typical mathematical proposition has the form A ⇒ B, where A is the assumption and B the conclusion. The proof of such a proposition consists of constructing1.1 Logical Symbolism3a chain A ⇒ C1 ⇒ · · · ⇒ Cn ⇒ B of implications, each element of which is eitheran axiom or a previously proved proposition.5In proofs we shall adhere to the classical rule of inference: if A is true andA ⇒ B, then B is also true.In proof by contradiction we shall also use the law of excluded middle, by virtueof which the statement A ∨ ¬A (A or not-A) is considered true independently of thespecific content of the statement A.
Consequently we simultaneously accept that¬(¬A) ⇔ A, that is, double negation is equivalent to the original statement.1.1.3 Some Special NotationFor the reader’s convenience and to shorten the writing, we shall agree to denote theend of a proof by the symbol .We also agree, whenever convenient, to introduce definitions using the specialsymbol := (equality by definition), in which the colon is placed on the side of theobject being defined.For example, the notationbf (x) dx := lim σ (f ; P , ξ )λ(P )→0adefines the left-hand side in terms of the right-hand side, whose meaning is assumedto be known.Similarly, one can introduce abbreviations for expressions already defined.
Forexamplenf (ξi )Δxi =: σ (f ; P , ξ )i=1introduces the notation σ (f ; P , ξ ) for the sum of special form on the left-hand side.1.1.4 Concluding RemarksWe note that here we have spoken essentially about notation only, without analyzingthe formalism of logical deductions and without touching on the profound questionsof truth, provability, and deducibility, which form the subject matter of mathematicallogic.How are we to construct mathematical analysis if we have no formalization oflogic? There may be some consolation in the fact that we always know more than5 Thenotation A ⇒ B ⇒ C will be used as an abbreviation for (A ⇒ B) ∧ (B ⇒ C).41 Some General Mathematical Concepts and Notationwe can formalize at any given time, or perhaps we should say we know how to domore than we can formalize.
This last sentence may be clarified by the well-knownproverb of the centipede who forgot how to walk when asked to explain exactly howit dealt with so many legs.The experience of all the sciences convinces us that what was considered clearor simple and unanalyzable yesterday may be subjected to reexamination or mademore precise today. Such was the case (and will undoubtedly be the case again)with many concepts of mathematical analysis, the most important theorems andmachinery of which were discovered in the seventeenth and eighteenth centuries,but which acquired its modern formalized form with a unique interpretation that isprobably responsible for its being generally accessible, only after the creation of thetheory of limits and the fully developed theory of real numbers needed for it in thenineteenth century.This is the level of the theory of real numbers from which we shall begin toconstruct the whole edifice of analysis in Chap.
2.As already noted in the preface, those who wish to make a rapid acquaintancewith the basic concepts and effective machinery of differential and integral calculusproper may begin immediately with Chap. 3, turning to particular places in the firsttwo chapters only as needed.1.1.5 ExercisesWe shall denote true assertions by the symbol 1 and false ones by 0. Then to eachof the statements ¬A, A ∧ B, A ∨ B, and A ⇒ B one can associate a so-called truthtable, which indicates its truth or falsehood depending on the truth of the statementsA and B.
These tables are a formal definition of the logical operations ¬, ∧, ∨, ⇒.Here they are:1. Check whether all of these tables agree with your concept of the correspondinglogical operation. (In particular, pay attention to the fact that if A is false, then theimplication A ⇒ B is always true.)1.2 Sets and Elementary Operations on Them52. Show that the following simple, but very useful relations, which are widely usedin mathematical reasoning, are true:a)b)c)d)e)¬(A ∧ B) ⇔ ¬A ∨ ¬B;¬(A ∨ B) ⇔ ¬A ∧ ¬B;(A ⇒ B) ⇔ (¬B ⇒ ¬A);(A ⇒ B) ⇔ (¬A ∨ B);¬(A ⇒ B) ⇔ A ∧ ¬B.1.2 Sets and Elementary Operations on Them1.2.1 The Concept of a SetSince the late nineteenth and early twentieth centuries the most universal languageof mathematics has been the language of set theory.
This is even manifest in one ofthe definitions of mathematics as the science that studies different structures (relations) on sets.6“We take a set to be an assemblage of definite, perfectly distinguishable objectsof our intuition or our thought into a coherent whole.” Thus did Georg Cantor,7 thecreator of set theory, describe the concept of a set.Cantor’s description cannot, of course, be considered a definition, since it appealsto concepts that may be more complicated than the concept of a set itself (and in anycase, have not been defined previously). The purpose of this description is to explainthe concept by connecting it with other concepts.The basic assumptions of Cantorian (or, as it is generally called, “naive”) settheory reduce to the following statements.10 .
A set may consist of any distinguishable objects.20 . A set is unambiguously determined by the collection of objects that compriseit.30 . Any property defines the set of objects having that property.If x is an object, P is a property, and P (x) denotes the assertion that x hasproperty P , then the class of objects having the property P is denoted {x | P (x)}.The objects that constitute a class or set are called the elements of the class or set.The set consisting of the elements x1 , .
. . , xn is usually denoted {x1 , . . . , xn }.Wherever no confusion can arise we allow ourselves to denote the one-element set{a} simply as a.The words “class”, “family”, “totality”, and “collection” are used as synonymsfor “set” in naive set theory.6 Bourbaki,N.
“The architecture of mathematics” in: N. Bourbaki, Elements of the history of mathematics, translated from the French by John Meldrum, Springer, New York, 1994.7 G. Cantor (1845–1918) – German mathematician, the creator of the theory of infinite sets and theprogenitor of set-theoretic language in mathematics.61 Some General Mathematical Concepts and NotationThe following examples illustrate the application of this terminology:––––––––the set of letters “a” occurring in the word “I”;the set of wives of Adam;the collection of ten decimal digits;the family of beans;the set of grains of sand on the Earth;the totality of points of a plane equidistant from two given points of the plane;the family of sets;the set of all sets.The variety in the possible degree of determinacy in the definition of a set leadsone to think that a set is, after all, not such a simple and harmless concept.And in fact the concept of the set of all sets, for example, is simply contradictory.Proof Indeed, suppose that for a set M the notation P (M) means that M is not anelement of itself.Consider the class K = {M | P (M)} of sets having property P .If K is a set either P (K) or ¬P (K) is true.
However, this dichotomy does notapply to K. Indeed, P (K) is impossible; for it would then follow from the definitionof K that K contains K as an element, that is, that ¬P (K) is true; on the other hand,¬P (K) is also impossible, since that means that K contains K as an element, whichcontradicts the definition of K as the class of sets that do not contain themselves aselements.Consequently K is not a set.This is the classical paradox of Russell,8 one of the paradoxes to which the naiveconception of a set leads.In modern mathematical logic the concept of a set has been subjected to detailedanalysis (with good reason, as we see).
However, we shall not go into that analysis. We note only that in the current axiomatic set theories a set is defined as amathematical object having a definite collection of properties.The description of these properties constitutes an axiom system. The core ofaxiomatic set theory is the postulation of rules by which new sets can be formedfrom given ones.
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