1610912322-b551b095a53deaf3d3fbd1ed05ae9b84 (824701), страница 7
Текст из файла (страница 7)
In general any of the current axiom systems is such that, on theone hand, it eliminates the known contradictions of the naive theory, and on theother hand it provides freedom to operate with specific sets that arise in differentareas of mathematics, most of all, in mathematical analysis understood in the broadsense of the word.Having confined ourselves for the time being to remarks on the concept of aset, we pass to the description of the set-theoretic relations and operations mostcommonly used in analysis.Those wishing a more detailed acquaintance with the concept of a set shouldstudy Sect.
1.4.2 in the present chapter or turn to the specialized literature.8 B.Russell (1872–1970) – British logician, philosopher, sociologist and social activist.1.2 Sets and Elementary Operations on Them7Fig. 1.11.2.2 The Inclusion RelationAs has already been pointed out, the objects that comprise a set are usually called theelements of the set. We tend to denote sets by uppercase letters and their elementsby the corresponding lowercase letters.The statement, “x is an element of the set X” is written briefly asx∈X(or X x),x∈/X(or X x).and its negation asWhen statements about sets are written, frequent use is made of the logical operators ∃ (“there exists” or “there are”) and ∀ (“every” or “for any”) which are calledthe existence and generalization quantifiers respectively.For example, the string ∀x ((x ∈ A) ⇔ (x ∈ B)) means that for any object x therelations x ∈ A and x ∈ B are equivalent.
Since a set is completely determined byits elements, this statement is usually written briefly asA = B,read “A equals B”, and means that the sets A and B are the same.Thus two sets are equal if they consist of the same elements.The negation of equality is usually written as A = B.If every element of A is an element of B, we write A ⊂ B or B ⊃ A and saythat A is a subset of B or that B contains A or that B includes A. In this connection the relation A ⊂ B between sets A and B is called the inclusion relation(Fig.
1.1).Thus(A ⊂ B) := ∀x (x ∈ A) ⇒ (x ∈ B) .If A ⊂ B and A = B, we shall say that the inclusion A ⊂ B is strict or that A isa proper subset of B.Using these definitions, we can now conclude that(A = B) ⇔ (A ⊂ B) ∧ (B ⊂ A).81 Some General Mathematical Concepts and NotationFig. 1.2If M is a set, any property P distinguishes in M the subsetx ∈ M | P (x)consisting of the elements of M that have the property.For example, it is obvious thatM = {x ∈ M | x ∈ M}.On the other hand, if P is taken as a property that no element of the set M has, forexample, P (x) := (x = x), we obtain the set∅ = {x ∈ M | x = x},called the empty subset of M.1.2.3 Elementary Operations on SetsLet A and B be subsets of a set M.a.
The union of A and B is the setA ∪ B := x ∈ M | (x ∈ A) ∨ (x ∈ B) ,consisting of precisely the elements of M that belong to at least one of the sets Aand B (Fig. 1.2).b. The intersection of A and B is the setA ∩ B := x ∈ M | (x ∈ A) ∧ (x ∈ B) ,formed by the elements of M that belong to both sets A and B (Fig. 1.3).c. The difference between A and B is the setA\B := x ∈ M | (x ∈ A) ∧ (x ∈/ B) ,consisting of the elements of A that do not belong to B (Fig. 1.4).1.2 Sets and Elementary Operations on Them9Fig. 1.3Fig. 1.4Fig. 1.5The difference between the set M and one of its subsets A is usually calledthe complement of A in M and denoted CM A, or CA when the set in which thecomplement of A is being taken is clear from the context (Fig.
1.5).Example As an illustration of the interaction of the concepts just introduced, let usverify the following relations (the so-called de Morgan9 rules):CM (A ∪ B) = CM A ∩ CM B,(1.1)CM (A ∩ B) = CM A ∪ CM B.(1.2)Proof We shall prove the first of these equalities by way of example: / (A ∪ B) ⇒ (x ∈/ A) ∧ (x ∈/ B) ⇒x ∈ CM (A ∪ B) ⇒ x ∈⇒ (x ∈ CM A) ∧ (x ∈ CM B) ⇒ x ∈ (CM A ∩ CM B) .Thus we have established thatCM (A ∪ B) ⊂ (CM A ∩ CM B) .9 A.de Morgan (1806–1871) – Scottish mathematician.(1.3)101 Some General Mathematical Concepts and NotationOn the other hand, x ∈ (CM A ∩ CM B) ⇒ (x ∈ CM A) ∧ (x ∈ CM B) ⇒ ⇒ (x ∈/ A) ∧ (x ∈/ B) ⇒ x ∈/ (A ∪ B) ⇒⇒ x ∈ CM (A ∪ B) ,that is,(CM A ∩ CM B) ⊂ CM (A ∪ B).Equation (1.1) follows from (1.3) and (1.4).(1.4)d.
The direct (Cartesian) product of sets. For any two sets A and B one can form anew set, namely the pair {A, B} = {B, A}, which consists of the sets A and B andno others. This set has two elements if A = B and one element if A = B.This set is called the unordered pair of sets A and B, to be distinguished from theordered pair (A, B) in which the elements are endowed with additional propertiesto distinguish the first and second elements of the pair {A, B}. The equality(A, B) = (C, D)between two ordered pairs means by definition that A = C and B = D. In particular,if A = B, then (A, B) = (B, A).Now let X and Y be arbitrary sets.
The setX × Y := (x, y) | (x ∈ X) ∧ (y ∈ Y ) ,formed by the ordered pairs (x, y) whose first element belongs to X and whosesecond element belongs to Y , is called the direct or Cartesian product of the sets Xand Y (in that order!).It follows obviously from the definition of the direct product and the remarksmade above about the ordered pair that in general X × Y = Y × X.
Equality holdsonly if X = Y . In this last case we abbreviate X × X as X 2 .The direct product is also called the Cartesian product in honor of Descartes,10who arrived at the language of analytic geometry in terms of a system of coordinates independently of Fermat.11 The familiar system of Cartesian coordinates inthe plane makes this plane precisely into the direct product of two real axes. Thisfamiliar object shows vividly why the Cartesian product depends on the order of thefactors. For example, different points of the plane correspond to the pairs (0, 1) and(1, 0).10 R. Descartes (1596–1650) – outstanding French philosopher, mathematician and physicist whomade fundamental contributions to scientific thought and knowledge.11 P. Fermat (1601–1665) – remarkable French mathematician, a lawyer by profession.
He wasone of the founders of a number of areas of modern mathematics: analysis, analytic geometry,probability theory, and number theory.1.2 Sets and Elementary Operations on Them11In the ordered pair z = (x1 , x2 ), which is an element of the direct product Z =X1 × X2 of the sets X1 and X2 , the element x1 is called the first projection of thepair z and denoted pr1 z, while the element x2 is the second projection of z and isdenoted pr2 z.By analogy with the terminology of analytic geometry, the projections of an ordered pair are often called the (first and second) coordinates of the pair.1.2.4 ExercisesIn Exercises 1, 2, and 3 the letters A, B, and C denote subsets of a set M.1.
Verify the following relations.a)b)c)d)e)(A ⊂ C) ∧ (B ⊂ C) ⇔ ((A ∪ B) ⊂ C);(C ⊂ A) ∧ (C ⊂ B) ⇔ (C ⊂ (A ∩ B));CM (CM A) = A;(A ⊂ CM B) ⇔ (B ⊂ CM A);(A ⊂ B) ⇔ (CM A ⊃ CM B).2. Prove the following statements.a)b)c)d)A ∪ (B ∪ C) = (A ∪ B) ∪ C =: A ∪ B ∪ C;A ∩ (B ∩ C) = (A ∩ B) ∩ C =: A ∩ B ∩ C;A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C);A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).3. Verify the connection (duality) between the operations of union and intersection:a) CM (A ∪ B) = CM A ∩ CM B;b) CM (A ∩ B) = CM A ∪ CM B.4.
Give geometric representations of the following Cartesian products.a)b)c)d)e)f)The product of two line segments (a rectangle).The product of two lines (a plane).The product of a line and a circle (an infinite cylindrical surface).The product of a line and a disk (an infinite solid cylinder).The product of two circles (a torus).The product of a circle and a disk (a solid torus).5. The set Δ = {(x1 , x2 ) ∈ X 2 | x1 = x2 } is called the diagonal of the Cartesiansquare X 2 of the set X. Give geometric representations of the diagonals of the setsobtained in parts a), b), and e) of Exercise 4.6.
Show thata) (X × Y = ∅) ⇔ (X = ∅) ∨ (Y = ∅), and if X × Y = ∅, thenb) (A × B ⊂ X × Y ) ⇔ (A ⊂ X) ∧ (B ⊂ Y ),c) (X × Y ) ∪ (Z × Y ) = (X ∪ Z) × Y ,121 Some General Mathematical Concepts and Notationd) (X × Y ) ∩ (X × Y ) = (X ∩ X ) × (Y ∩ Y ).Here ∅ denotes the empty set, that is, the set having no elements.7.
By comparing the relations of Exercise 3 with relations a) and b) from Exercise 2of Sect. 1.1, establish a correspondence between the logical operators ¬, ∧, ∨ andthe operations C, ∩, and ∪ on sets.1.3 Functions1.3.1 The Concept of a Function (Mapping)We shall now describe the concept of a functional relation, which is fundamentalboth in mathematics and elsewhere.Let X and Y be certain sets. We say that there is a function defined on X withvalues in Y if, by virtue of some rule f , to each element x ∈ X there correspondsan element y ∈ Y .In this case the set X is called the domain of definition of the function.
Thesymbol x used to denote a general element of the domain is called the argumentof the function, or the independent variable. The element y0 ∈ Y corresponding toa particular value x0 ∈ X of the argument x is called the value of the function atx0 , or the value of the function at the value x = x0 of its argument, and is denotedf (x0 ). As the argument x ∈ X varies, the value y = f (x) ∈ Y , in general, variesdepending on the values of x. For that reason, the quantity y = f (x) is often calledthe dependent variable.The setf (X) := y ∈ Y | ∃x (x ∈ X) ∧ y = f (x)of values assumed by a function on elements of the set X will be called the set ofvalues or the range of the function.The term “function” has a variety of useful synonyms in different areas of mathematics, depending on the nature of the sets X and Y : mapping, transformation,morphism, operator, functional.
The commonest is mapping, and we shall also useit frequently.For a function (mapping) the following notations are standard:f : X → Y,fX −→ Y.When it is clear from the context what the domain and range of a function are,one also uses the notation x → f (x) or y = f (x), but more frequently a function ingeneral is simply denoted by the single symbol f .Two functions f1 and f2 are considered identical or equal if they have the samedomain X and at each element x ∈ X the values f1 (x) and f2 (x) are the same. Inthis case we write f1 = f2 .1.3 Functions13If A ⊂ X and f : X → Y is a function, we denote by f |A or f |A the functionϕ : A → Y that agrees with f on A.
More precisely, f |A (x) := ϕ(x) if x ∈ A. Thefunction f |A is called the restriction of f to A, and the function f : X → Y is calledan extension or a continuation of ϕ to X.We see that it is sometimes necessary to consider a function ϕ : A → Y definedon a subset A of some set X while the range ϕ(A) of ϕ may also turn out be a subsetof Y that is different from Y . In this connection, we sometimes use the term domainof departure of the function to denote any set X containing the domain of a function,and domain of arrival to denote any subset of Y containing its range.Thus, defining a function (mapping) involves specifying a triple (X, Y, f ), whereX is the set being mapped, or domain of the function;Y is the set into which the mapping goes, or a domain of arrival of the function;f is the rule according to which a definite element y ∈ Y is assigned to eachelement x ∈ X.The asymmetry between X and Y that appears here reflects the fact that themapping goes from X to Y , and not the other direction.Now let us consider some examples of functions.Example 1 The formulas l = 2πr and V = 43 πr 3 establish functional relationshipsbetween the circumference l of a circle and its radius r and between the volume Vof a ball and its radius r.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
