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This none other than the successivecomputation of f n (x0 ), where f (x) = 12 (x + xa ). Such a procedure, in which thevalue of the function computed at the each step becomes its argument at the nextstep, is called a recursive procedure. Recursive procedures are widely used in mathematics.We further note that even when both compositions g ◦ f and f ◦ g are defined,in generalg ◦ f = f ◦ g.Indeed, let us take for example the two-element set {a, b} and the mappings f :{a, b} → a and g : {a, b} → b. Then it is obvious that g ◦ f : {a, b} → b whilef ◦ g : {a, b} → a.The mapping f : X → X that assigns to each element of X the element itself,fthat is x −→ y, will be denoted eX and called the identity mapping on X.Lemma(g ◦ f = eX ) ⇒ (g is surjective) ∧ (f is injective).Proof Indeed, if f : X → Y, g : Y → X, and g ◦ f = eX : X → X, thenX = eX (X) = (g ◦ f )(X) = g f (X) ⊂ g(Y )and hence g is surjective.Further, if x1 ∈ X and x2 ∈ X, then (x1 = x2 ) ⇒ eX (x1 ) = eX (x2 ) ⇒ (g ◦ f )(x1 ) = (g ◦ f )(x2 ) ⇒ ⇒ g f (x1 ) = g f (x2 ) ⇒ f (x1 ) = f (x2 ) ,and therefore f is injective.1.3 Functions19Using the operation of composition of mappings one can describe mutually inverse mappings.Proposition The mappings f : X → Y and g : Y → X are bijective and mutuallyinverse to each other if and only if g ◦ f = eX and f ◦ g = eY .Proof By the lemma the simultaneous fulfillment of the conditions g ◦ f = eX andf ◦ g = eY guarantees the surjectivity and injectivity, that is, the bijectivity, of bothmappings.These same conditions show that y = f (x) if and only if x = g(y).In the preceding discussion we started with an explicit construction of the inversemapping.
It follows from the proposition just proved that we could have given a lessintuitive, yet more symmetric definition of mutually inverse mappings as those mappings that satisfy the two conditions g ◦ f = eX and f ◦ g = eY . (In this connection,see Exercise 6 at the end of this section.)1.3.4 Functions as Relations. The Graph of a FunctionIn conclusion we return once again to the concept of a function. We note that it hasundergone a lengthy and rather complicated evolution.The term function first appeared in the years from 1673 to 1692 in works ofG. Leibniz (in a somewhat narrower sense, to be sure).
By the year 1698 the term hadbecome established in a sense close to the modern one through the correspondencebetween Leibniz and Johann Bernoulli.13 (The letter of Bernoulli usually cited inthis regard dates to that same year.)Many great mathematicians have participated in the formation of the modernconcept of functional dependence.A description of a function that is nearly identical to the one given at the beginning of this section can be found as early as the work of Euler (mid-eighteenthcentury) who also introduced the notation f (x).
By the early nineteenth century ithad appeared in the textbooks of S. Lacroix.14 A vigorous advocate of this conceptof a function was N.I. Lobachevskii,15 who noted that “a comprehensive view of13 Johann Bernoulli (1667–1748) – one of the early representatives of the distinguished Bernoullifamily of Swiss scholars; he studied analysis, geometry and mechanics. He was one of the foundersof the calculus of variations.
He gave the first systematic exposition of the differential and integralcalculus.14 S.F. Lacroix (1765–1843)– French mathematician and educator (professor at the École Normaleand the École Polytechnique, and member of the Paris Academy of Sciences).15 N.I. Lobachevskii (1792–1856) – great Russian scholar, to whom belongs the credit – shared withthe great German scientist C.F. Gauss (1777–1855) and the outstanding Hungarian mathematicianJ.
Bólyai (1802–1860) – for having discovered the non-Euclidean geometry that bears his name.201 Some General Mathematical Concepts and Notationtheory admits only dependence relationships in which the numbers connected witheach other are understood as if they were given as a single unit.”16 It is this idea ofprecise definition of the concept of a function that we are about to explain.The description of the concept of a function given at the beginning of this sectionis quite dynamic and reflects the essence of the matter. However, by modern canonsof rigor it cannot be called a definition, since it uses the concept of a correspondence,which is equivalent to the concept of a function. For the reader’s information weshall show here how the definition of a function can be given in the language of settheory. (It is interesting that the concept of a relation, to which we are now turning,preceded the concept of a function, even for Leibniz.)a.
RelationsDefinition 1 A relation R is any set of ordered pairs (x, y).The set X of first elements of the ordered pairs that constitute R is called thedomain of definition of R, and the set Y of second elements of these pairs the rangeof values of R.Thus, a relation can be interpreted as a subset R of the direct product X × Y . IfX ⊂ X and Y ⊂ Y , then of course R ⊂ X × Y ⊂ X × Y , so that a given relationcan be defined as a subset of different sets.Any set containing the domain of definition of a relation is called a domain ofdeparture for that relation.
A set containing the region of values is called a domainof arrival of the relation.Instead of writing (x, y) ∈ R, we often write xRy and say that x is connectedwith y by the relation R.If R ⊂ X 2 , we say that the relation R is defined on X.Let us consider some examples.Example 13 The diagonalΔ = (a, b) ∈ X 2 | a = bis a subset of X 2 defining the relation of equality between elements of X. Indeed,aΔb means that (a, b) ∈ Δ, that is, a = b.Example 14 Let X be the set of lines in a plane.Two lines a ∈ X and b ∈ X will be considered to be in the relation R, and weshall write aRb, if b is parallel to a. It is clear that this condition distinguishes a setR of pairs (a, b) in X 2 such that aRb.
It is known from geometry that the relationof parallelism between lines has the following properties:aRa (reflexivity);16 Lobachevskii, N.I. Complete Works, Vol. 5, Moscow–Leningrad: Gostekhizdat, 1951, p. 44 (Rus-sian).1.3 Functions21aRb ⇒ bRa (symmetry);(aRb) ∧ (bRc) ⇒ aRc (transitivity).A relation R having the three properties just listed, that is, reflexivity,17 symmetry, and transitivity, is usually called an equivalence relation. An equivalencerelation is denoted by the special symbol ∼, which in this case replaces the letter R.Thus, in the case of an equivalence relation we shall write a ∼ b instead of aRb andsay that a is equivalent to b.Example 15 Let M be a set and X = P(M) the set of its subsets. For two arbitraryelements a and b of X = P(M), that is, for two subsets a and b of M, one of thefollowing three possibilities always holds: a is contained in b; b is contained in a;a is not a subset of b and b is not a subset of a.As an example of a relation R on X 2 , consider the relation of inclusion for subsets of M, that is, make the definitionaRb := (a ⊂ b).This relation obviously has the following properties:aRa (reflexivity);(aRb) ∧ (bRc) ⇒ aRc (transitivity);(aRb) ∧ (bRa) ⇒ aΔb, that is, a = b (antisymmetry).A relation between pairs of elements of a set X having these three properties isusually called a partial ordering on X.
For a partial ordering relation on X, we oftenwrite a b and say that b follows a.If the condition∀a ∀b (aRb) ∨ (bRa)holds in addition to the last two properties defining a partial ordering relation, thatis, any two elements of X are comparable, the relation R is called an ordering, andthe set X with the ordering defined on it is said to be linearly ordered.The origin of this term comes from the intuitive image of the real line R on whicha relation a ≤ b holds between any pair of real numbers.b. Functions and Their GraphsA relation R is said to be functional if(xRy1 ) ∧ (xRy2 ) ⇒ (y1 = y2 ).17 Forthe sake of completeness it is useful to note that a relation R is reflexive if its domain ofdefinition and its range of values are the same and the relation a Ra holds for any element a in thedomain of R.221 Some General Mathematical Concepts and NotationA functional relation is called a function.In particular, if X and Y are two sets, not necessarily distinct, a relation R ⊂X × Y between elements x of X and y of Y is a functional relation on X if for everyx ∈ X there exists a unique element y ∈ Y in the given relation to x, that is, suchthat xRy holds.Such a functional relation R ⊂ X × Y is a mapping from X into Y , or a functionfrom X into Y .We shall usually denote functions by the letter f .
If f is a function, we shallfwrite y = f (x) or x −→ y, as before, rather than xfy, calling y = f (x) the valueof f at x or the image of x under f .As we now see, assigning an element y ∈ Y “corresponding” to x ∈ X in accordance with the “rule” f , as was discussed in the original description of the conceptof a function, amounts to exhibiting for each x ∈ X the unique y ∈ Y such that xfy,that is, (x, y) ∈ f ⊂ X × Y .The graph of a function f : X → Y , as understood in the original description, isthe subset Γ of the direct product X × Y whose elements have the form (x, f (x)).ThusΓ := (x, y) ∈ X × Y | y = f (x) .In the new description of the concept of a function, in which we define it as asubset f ⊂ X × Y , of course, there is no longer any difference between a functionand its graph.We have exhibited the theoretical possibility of giving a formal set-theoretic definition of a function, which reduces essentially to identifying a function and its graph.However, we do not intend to confine ourselves to that way of defining a function.At times it is convenient to define a functional relation analytically, at other timesby giving a table of values, and at still other times by giving a verbal description of aprocess (algorithm) making it possible to find the element y ∈ Y corresponding to agiven x ∈ X.
With each method of presenting a function it is meaningful to ask howthe function could have been defined using its graph. This problem can be stated asthe problem of constructing the graph of the function. Defining numerical-valuedfunctions by a good graphical representation is often useful because it makes thebasic qualitative properties of the functional relation visualizable. One can also usegraphs (nomograms) for computations; but, as a rule, only in cases where high precision is not required.
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