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For precise computations we do use the table definition of afunction, but more often we use an algorithmic definition that can be implementedon a computer.1.3.5 Exercises1. The composition R2 ◦ R1 of the relations R1 and R2 is defined as follows:R2 ◦ R1 := (x, z) | ∃y (xR1 y ∧ yR2 z) .1.3 Functions23In particular, if R1 ⊂ X × Y and R2 ⊂ Y × Z, then R = R2 ◦ R1 ⊂ X × Z, andxRz := ∃y (y ∈ Y ) ∧ (xR1 y) ∧ (yR2 z) .a) Let ΔX be the diagonal of X 2 and ΔY the diagonal of Y 2 .
Show that if therelations R1 ⊂ X × Y and R2 ⊂ Y × X are such that (R2 ◦ R1 = ΔX ) ∧ (R1 ◦ R2 =ΔY ), then both relations are functional and define mutually inverse mappings of Xand Y .b) Let R ⊂ X 2 . Show that the condition of transitivity of the relation R is equivalent to the condition R ◦ R ⊂ R.c) The relation R ⊂ Y × X is called the transpose of the relation R ⊂ X × Y if(yR x) ⇔ (xRy).Show that a relation R ⊂ X 2 is antisymmetric if and only if R ∩ R ⊂ ΔX .d) Verify that any two elements of X are connected (in some order) by the relation R ⊂ X 2 if and only if R ∪ R = X 2 .2.
Let f : X → Y be a mapping. The pre-image f −1 (y) ⊂ X of the element y ∈ Yis called the fiber over y.a) Find the fibers for the following mappings:pr1 : X1 × X2 → X1 ,pr2 : X1 × X2 → X2 .b) An element x1 ∈ X will be considered to be connected with an element x2 ∈X by the relation R ⊂ X 2 , and we shall write x1 Rx2 if f (x1 ) = f (x2 ), that is, x1and x2 both lie in the same fiber.Verify that R is an equivalence relation.c) Show that the fibers of a mapping f : X → Y do not intersect one anotherand that the union of all the fibers is the whole set X.d) Verify that any equivalence relation between elements of a set makes it possible to represent the set as a union of mutually disjoint equivalence classes of elements.3.
Let f : X → Y be a mapping from X into Y . Show that if A and B are subsetsof X, thena)b)c)d)(A ⊂ B) ⇒ (f (A) ⊂ f (B)) = (A ⊂ B).(A = ∅) ⇒ (f (A) = ∅),f (A ∩ B) ⊂ f (A) ∩ f (B),f (A ∪ B) = f (A) ∪ f (B);if A and B are subsets of Y , thene) (A ⊂ B ) ⇒ (f −1 (A ) ⊂ f −1 (B )),f) f −1 (A ∩ B ) = f −1 (A ) ∩ f −1 (B ),g) f −1 (A ∪ B ) = f −1 (A ) ∪ f −1 (B );if Y ⊃ A ⊃ B , thenh) f −1 (A \B ) = f −1 (A )\f −1 (B ),241 Some General Mathematical Concepts and Notationi) f −1 (CY A ) = CX f −1 (A );and for any A ⊂ X and B ⊂ Yj) f −1 (f (A)) ⊃ A,k) f (f −1 (B )) ⊂ B .4. Show that the mapping f : X → Y isa) surjective if and only if f (f −1 (B )) = B for every set B ⊂ Y ;b) bijective if and only if −1 f (A) = A ∧ f f −1 B = B ffor every set A ⊂ X and every set B ⊂ Y .5.
Verify that the following statements about a mapping f : X → Y are equivalent:a)b)c)d)e)f is injective;f −1 (f (A)) = A for every A ⊂ X;f (A ∩ B) = f (A) ∩ f (B) for any two subsets A and B of X;f (A) ∩ f (B) = ∅ ⇔ A ∩ B = ∅;f (A\B) = f (A)\f (B) whenever X ⊃ A ⊃ B.6. a) If the mappings f : X → Y and g : Y → X are such that g ◦ f = eX , whereeX is the identity mapping on X, then g is called a left inverse of f and f a rightinverse of g. Show that, in contrast to the uniqueness of the inverse mapping, theremay exist many one-sided inverse mappings.Consider, for example, the mappings f : X → Y and g : Y → X, where X is aone-element set and Y a two-element set, or the mappings of sequences given byfa(x1 , .
. . , xn , . . .) −→ (a, x1 , . . . , xn , . . .),g(y2 , . . . , yn , . . .) ←−[ (y1 , y2 , . . . , yn , . . .).b) Let f : X → Y and g : Y → Z be bijective mappings. Show that the mappingg ◦ f : X → Z is bijective and that (g ◦ f )−1 = f −1 ◦ g −1 .c) Show that the equality(g ◦ f )−1 (C) = f −1 g −1 (C)holds for any mappings f : X → Y and g : Y → Z and any set C ⊂ Z.d) Verify that the mapping F : X × Y → Y × X defined by the correspondence(x, y) → (y, x) is bijective.
Describe the connection between the graphs of mutuallyinverse mappings f : X → Y and f −1 : Y → X.7. a) Show that for any mapping f : X → Y the mapping F : X → X × Y definedFby the correspondence x −→(x, f (x)) is injective.b) Suppose a particle is moving at uniform speed on a circle Y ; let X be theftime axis and x −→ y the correspondence between the time x ∈ X and the position1.4 Supplementary Material25y = f (x) ∈ Y of the particle. Describe the graph of the function f : X → Y inX × Y.8. a) For each of the examples 1–12 considered in Sect. 1.3 determine whether themapping defined in the example is surjective, injective, or bijective or whether itbelongs to none of these classes.b) Ohm’s law I = V /R connects the current I in a conductor with the potentialdifference V at the ends of the conductor and the resistance R of the conductor.Give sets X and Y for which some mapping O : X → Y corresponds to Ohm’s law.What set is the relation corresponding to Ohm’s law a subset of?c) Find the mappings G−1 and L−1 inverse to the Galilean and Lorentz transformations.9.
a) A set S ⊂ X is stable with respect to a mapping f : X → X if f (S) ⊂ S.Describe the sets that are stable with respect to a shift of the plane by a given vectorlying in the plane.b) A set I ⊂ X is invariant with respect to a mapping f : X → X if f (I ) = I .Describe the sets that are invariant with respect to rotation of the plane about a fixedpoint.c) A point p ∈ X is a fixed point of a mapping f : X → X if f (p) = p.
Verifythat any composition of a shift, a rotation, and a similarity transformation of theplane has a fixed point, provided the coefficient of the similarity transformation isless than 1.d) Regarding the Galilean and Lorentz transformations as mappings of the planeinto itself for which the point with coordinates (x, t) maps to the point with coordinates (x , t ), find the invariant sets of these transformations.10.
Consider the steady flow of a fluid (that is, the velocity at each point of the flowdoes not change over time). In time t a particle at point x of the flow will moveto some new point ft (x) of space. The mapping x → ft (x) that arises thereby onthe points of space occupied by the flow depends on time and is called the mappingafter time t. Show that ft2 ◦ ft1 = ft1 ◦ ft2 = ft1 +t2 and ft ◦ f−t = eX .1.4 Supplementary Material1.4.1 The Cardinality of a Set (Cardinal Numbers)The set X is said to be equipollent to the set Y if there exists a bijective mappingof X onto Y , that is, a point y ∈ Y is assigned to each x ∈ X, the elements of Yassigned to different elements of X are different, and every point of Y is assigned tosome point of X.Speaking fancifully, each element x ∈ X has a seat all to itself in Y , and there areno vacant seats y ∈ Y .261 Some General Mathematical Concepts and NotationIt is clear that the relation XRY thereby introduced is an equivalence relation.For that reason we shall write X ∼ Y instead of XRY , in accordance with our earlierconvention.The relation of equipollence partitions the collection of all sets into classes ofmutually equivalent sets.
The sets of an equivalence class have the same number ofelements (they are equipollent), and sets from different equivalence classes do not.The class to which a set X belongs is called the cardinality of X, and also thecardinal or cardinal number of X. It is denoted card X.
If X ∼ Y , we write cardX = card Y .The idea behind this construction is that it makes possible a comparison ofthe numbers of elements in sets without resorting to an intermediate count, thatis, without measuring the number by comparing it with the natural numbers N ={1, 2, 3, . . .}. Doing the latter, as we shall soon see, is sometimes not even theoretically possible.The cardinal number of a set X is said to be not larger than the cardinal numberof a set Y , and we write card X ≤ card Y , if X is equipollent to some subset of Y .Thus,(card X ≤ card Y ) := ∃Z ⊂ Y (card X = card Z).If X ⊂ Y , it is clear that card X ≤ card Y . It turns out, however, that the relationX ⊂ Y does not exclude the inequality card Y ≤ card X, even when X is a propersubset of Y .xis a bijective mapping of the intervalFor example, the correspondence x → 1−|x|−1 < x < 1 of the real axis R onto the entire axis.The possibility of being equipollent to a proper subset of itself is a characteristicof infinite sets that Dedekind18 even suggested taking as the definition of an infiniteset.
Thus a set is called finite (in the sense of Dedekind) if it is not equipollent toany proper subset of itself; otherwise, it is called infinite.Just as the relation of inequality orders the real numbers on a line, the inequalityjust introduced orders the cardinal numbers of sets. To be specific, one can provethat the relation just constructed has the following properties:10 (card X ≤ card Y ) ∧ (card Y ≤ card Z) ⇒ (card X ≤ card Z) (obvious).20 (card X ≤ card Y ) ∧ (card Y ≤ card X) ⇒ (card X = card Y ) (the Schröder–Bernstein theorem19 ).03 ∀X ∀Y (card X ≤ card Y ) ∨ (card Y ≤ card X) (Cantor’s theorem).Thus the class of cardinal numbers is linearly ordered.18 R. Dedekind (1831–1916) – German algebraist who took an active part in the development of thetheory of a real number.
He was the first to propose the axiomatization of the set of natural numbersusually called the Peano axiom system after G. Peano (1858–1932), the Italian mathematician whoformulated it somewhat later.19 F. Bernstein (1878–1956) – German mathematician, a student of G. Cantor. E. Schröder (1841–1902) – German mathematician.1.4 Supplementary Material27We say that the cardinality of X is less than the cardinality of Y and writecard X < card Y , if card X ≤ card Y but card X = card Y .
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