1)Definition. A correspondence in which each of the elements of the set X is associated with a single element from the set Y is called display.

3) If the element x corresponds y, then y called element image x, but x -element preimage y. Write: or y = f(x). Lots of A of all elements having the same image is called full preimage of an element y.

4) Function scope are all values ​​of x for which the function exists. In other words, the scope of the function given by the formula is all the values ​​of the argument, except for those that lead to actions that we cannot perform. At the moment we know only two such actions. We cannot divide by zero, and we cannot take the square root of a negative number.

5)Ways of setting, types and properties of mappings

Setting methods

EXPRESSION or FORMULA. The variable to replace with an element from the scope is called a function argument. This explicitly indicates the procedure for calculating the value f(x) of the function f on the argument x, more precisely, for any value of the argument. In fact, in this way, we specify the rule for calculating the value of the function f for an arbitrary value of the argument x. TABLE. The table of function values ​​usually consists of two lines. The first line lists all (!) elements of the scope, and the second line lists the function values ​​corresponding to them.

SCHEDULE. The graph of the function f is the set of points in the plane with coordinates x, f(x) .

ALGORITHM. X→|A|→y=y(x)

6)Operations on mappings

1. Reversal y:A→B Y(x)=y

2. Composition of mappings

Y1:A→B y2:B→c

Composition y1*y2 mapping y1:a->c such that y(x)=y1*y2(x)=Z( E yϵB)(y1=y1(x)&y2(y)=Z)

7) Functions as a special class of mappings

8) Classification of functions by type of plural

3. Binary relations

1) Attitude

2) binary relation is a binomial relation between any two sets A and B, i.e. any subset of the Cartesian product of these sets:    A B.

3) examples Examples of binary relations:

4) Ways of setting

5) sv-va binary relations

6) Element projection(a, b) from the set Ax B to the set A is an element a. Similarly, the element b is the projection of the element (a, b) of the set Ax B onto the set B. The projection of the set EAx B onto A is the set of all those elements from A that are projections of elements from E onto the set A

7) Slice of a binary relation. Distinguish between a slice of a binary relation through an element and through a subset of the first basic set.

8) Factorials

9) Equivalence relation

10) connection with partitions

11) binary relationť on set A(ť  AxA) called the relation t tolerance if it is reflexive and symmetric.

12) its connection with the coating

13) order relation

14) str-ra ordered plurals

15) Lattice is a partially ordered set in which each two-element subset has both the best upper (sup) and the best lower (inf) faces. This implies the existence of these faces for any non-empty finite subsets. A lattice can also be defined as a universal algebra with two binary operations (they are denoted \/ and /\ or + and ∙)

Display. Injective, surjective and bijective mappings. Equivalent sets.

Let X, Y be arbitrary non-empty sets.
Definition. Display f from set X to set Y is a rule by which each element x∈X is assigned a uniquely defined element y∈Y.
The set X is called the domain of the mapping f; the set Y is its range.
Synonyms express the fact that f is a mapping from X to Y.

Element at∈Y, which, using the mapping f assigned to an element X∈X is called way element X and is denoted by f(x); in the same situation the element X called prototype element at. Full preimage of an element at we will call the set of all preimages at. It follows from the definition of a mapping that the complete pre-images of different elements have no common elements.

When the X range and the Y range of a given mapping f match, then f is called a transformation of the set X. If BUT is an arbitrary subset of the set X, then the set f(A) = {y|y = f(x) for some x∈BUT) is called the image of the set BUT when displayed f.
Image f(X) of the entire domain of definition of X is called the set of values ​​of the mapping f.
Often the scope and set of display values f denoted by D( f) and E( f) respectively.

Display f from X to Y is called injective, if for any x1, x2∈X from the inequality x1 ≠ x2 follows the inequality f(x1) ≠ f(x2).

Display f from X to Y is called surjective if the set of values f(X) is the same as the Y range.
If we use the concept of a complete preimage, then the definition can be formulated differently. Display f from X to Y is called surjective, if the full preimage of an arbitrary element y∈Y is a non-empty set.

Display f from X to Y is called bijective if it is surjective and injective at the same time.

If there is an injective (respectively bijective) mapping from X to Y, then we say that the cardinality of X is at most the cardinality of Y (respectively power X is equal to power Y).

Display

DISPLAY -I; cf. to Display - display and Display - display. O. maritime themes in painting. True, accurate, adequate about. Artistic, symbolic O. in the consciousness of the phenomena of reality.

display

(math.) sets X into the multitude Y X sets X y = f(X) sets Y, is called the image of the element X. For example, a geographical map can be viewed as the result of displaying the earth's surface (or part of it) onto a piece of a plane. The term "mapping" is equivalent to the term "function".

DISPLAY

DISPLAY (in mathematics) of a set X into the multitude Y, a correspondence due to which each element X sets X matches a specific element at=f(X) sets Y, called the image of the element X. For example, a geographic map can be viewed as the result of displaying the earth's surface (or part of it) onto a piece of a plane. The term "mapping" is equivalent to the term "function".

encyclopedic Dictionary. 2009 .

Synonyms:

See what "display" is in other dictionaries:

Display- conversion of the input data stream of the internal encoder into two output streams, which are in-phase and quadrature components, fed to the corresponding inputs of the modulator Source: OST 4 ... Dictionary-reference book of terms of normative and technical documentation

Representation, image, depiction, description, recreation, depiction, display; transformation, transformation, transformation; reproduction, transmission, reflection, indication, expression, delineation Dictionary of Russian synonyms. display 1. see… … Synonym dictionary

display- A logical relationship between a set of values ​​(for example, network addresses on one network) and objects in another set (for example, addresses on another network). mapping From the most general point of view, this is the rule according to which ... ...

MAPPING (in mathematics) of the set X to the set Y is a correspondence, due to which each element x of the set X corresponds to a certain element y \u003d f (x) of the set Y, called the image of the element x. For example, a geographic map can ... ... Big Encyclopedic Dictionary

DISPLAY, display, cf. 1. only units Action under ch. display display and display display. Display of reality. 2. What is displayed, the displayed phenomenon. 3. Same as reflection in 5 digits. (philosophical). Reflection theory ... ... Explanatory Dictionary of Ushakov

Display

Display- from the most general point of view, this is a rule according to which the elements of one set are assigned to the elements of another set. Therefore, it is sometimes said that a mapping is a tuple consisting of three elements: ... ... Economic and Mathematical Dictionary

DISPLAY, i, cf. 1. see display. 2. What is displayed is an image. True, accurate about. Explanatory dictionary of Ozhegov. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Explanatory dictionary of Ozhegov

display on- - [L.G. Sumenko. English Russian Dictionary of Information Technologies. M.: GP TsNIIS, 2003.] Topics information technology in general EN onto function ... Technical Translator's Handbook

A single-valued law, according to which each element of some given set X is associated with a well-defined element of another given set Y (in this case, X may coincide with Y). Such a relationship between elements and is written in ... ... Mathematical Encyclopedia

"Display" request redirects here. See also other meanings. This article provides a general definition of a mathematical function. In secondary schools and in non-mathematical specialties of higher educational institutions, they study a simpler ... ... Wikipedia

Books

• conformal mapping. , Carathéodory K.. Reproduced in the original author's spelling of the 1934 edition (ONTI publishing house) ...
• Transfer, processing, display of information. Collection of materials of the 26th All-Russian Scientific and Practical Conference, Collection of articles. This collection includes materials of the All-Russian scientific and practical conference "Transmission, processing, display of information", held in Krasnodar and in the village. Terskol,…

The function , where are the complex numbers satisfying the condition , is called fractional linear, and the mapping carried out by it - fractional linear display. For , we must assume that , , and for , we must assume that .

Exists the only linear-fractional function that maps given three different points of the extended complex plane to given three different points, respectively. It is found from the relation

which should be considered as an equation for . In this case, if some of the numbers are equal, then the fraction, in which the numerator and denominator is present, should be considered equal to 1. For example, if w 1 = , then it should be considered

The points and are called symmetrical about the circle, if they are located on the same ray coming from the center, and

The linear-fractional function maps a circle to a circle ( circular property), and the points that are symmetric with respect to the circle - into points that are symmetric with respect to the image of this circle ( symmetry property). Wherein the line should be considered as a circle passing through ∞ and closed at an infinitely distant point.

To find the image of an oriented circle (or a straight line) under a linear-fractional mapping, you need to take three different points on a given circle according to the direction of the bypass, find their images and draw a circle through them, which will be the image of this circle. The direction of the bypass on it must be taken from point to point and from to.

To find the image of a part of a circle or a straight line (arc, segment, ray) with a linear-fractional mapping, you need to take three points on it: initial, some kind of “middle” and final, find their images, draw a circle through them and take that part , for which is the starting point, is the “midpoint”, and is the end point.

To find the image of a region bounded by arcs of circles and parts of straight lines, one must choose the direction of the bypass on the boundary of the region so that the region remains on the left, and find the images of all parts of the boundary, taking into account their directions. These images together form a certain oriented closed line, perhaps unlimited, i.e. closed in . Then the region remaining to the left of this line will be the image of the original region.

To find any one conformal mapping of a region bounded by a circle (or a straight line) onto a similar region , one must choose the directions for bypassing the boundaries and regions and so that the regions remain on the left. Then, on the boundaries and, according to the directions of bypasses, take three different points and, accordingly, from equation (1) find a linear-fractional function , which will be one of the conformal mappings of the region onto the region .

In the general case, the conformal mapping of the unit circle onto the unit circle has the form:

the conformal mapping of the upper half-plane Im z > 0 onto the unit circle has the form:

the conformal mapping of the upper half-plane Im z > 0 onto the upper half-plane Im w > 0 has the form:

Tasks

1. Find a linear-fractional function that maps points to points respectively.

Solution: Substituting into relation (1) the given values

from where we find:

2 . Find a point symmetrical with a point about a circle.

Solution. From fig. 1, which shows the point z 1 = 3 and the circle, it can be seen that the desired symmetrical point is located inside the circle and has the form , where x > -2. This follows from the similarity of the corresponding triangles. Substituting z 1 , z 2 into the equality

we get: , whence, taking into account the inequality x > -2, we find . Then .

3. Find images of circles when displayed

Solution. Because

then the equations of circles have the form:

Substituting here found from the equation , we get:

Considering , we obtain a family of vertical lines

4. Find images of the region D when displaying if

Solution. a) Region D and the positive orientation of its boundary are shown in Fig. 2.

The boundary of the region in this case consists of two parts: a semicircle and two rays, which should be considered as one continuous part of the straight line Im z = 0, since the straight line is considered to be a circle passing through , i.e. continuous curve closed in . On these rays, as on one part of the boundary, we choose the starting point z 1 = -1, the middle point z 2 = , the end point z 3 = 1 and find their images

Let's draw a circle through the point - , 1 and take the part for which - - the beginning, 1 - the middle point, - the end. It will be the arc G 1 (Fig. 3). The direction of the bypass on the arc Г 1 is taken from - to 1 and from 1 to . This arc will be the image of the combination of two rays.

Find the image of a semicircle. The images of the beginning 1, the middle point - and the end -1 of the semicircle will be the points , 0 and - respectively. The circle passing through these points is a straight line Re w = 0, therefore, the image of the semicircle will be the segment Г 2 with ends and - directed from top to bottom (Fig. 3).

Consequently, the image of the boundary when displayed will be a closed curve Г 1 Г 2 directed counterclockwise, and the image of the region D will be the semicircle shown in Fig. 3.

b) In this case, the region D is an extended complex plane C with a cut along the segment [-2; 1] (Fig. 4).

Since the linear-fractional function maps to , then the image of the region D will be , from which the image of the segment [-2;1] should be thrown out. Since the images of the beginning -2, the “middle point” 0 and the end 1 during the display will be respectively the points , then the image of the segment [-2;1] will be the ray . Then the image of the region D will be a plane with a cut along the ray (Fig. 5).

c) The boundary of the region D consists of a straight line , oriented from left to right, and a circle , oriented counterclockwise (Fig. 6). When displayed, the points located on the line according to the direction of the bypass, respectively, go to the points. Hence, the line

goes into a straight line, oriented from right to left (Fig. 7). Similarly, taking the points 2 , 1+ , 0 on the circle and calculating their images , we find the image of the circle . It will be a straight line, oriented from left to right. This means that the image of the boundary will be a set of straight lines Г 1 and Г 2, and the image of the region D will be the strip shown in Fig. 7.

5. Find some conformal mapping of the region onto the half-plane.

Solution. Let's choose the directions of bypassing the boundaries of the regions D 1 and D 2 (Fig. 8) so that the regions remain on the left. According to these directions on the boundaries and take three points and , substituting them into equation (1), we find a linear-fractional mapping

which will be one of the desired conformal mappings.

6. Find a conformal mapping of the upper half-plane onto the unit circle that satisfies the conditions .

Solution. Since the general view of the conformal mapping of the upper half-plane onto the unit circle has the form

then the numbers must be chosen so that

whence = ,

Hence, the desired conformal mapping has the form

7. Find a conformal mapping of the half-plane Re z + Im z< 0 на круг удовлетворяющее условиям

Solution. Since any conformal mapping of a region bounded by a circle (or a line) onto a similar region is linear-fractional, then, according to the symmetry property of a linear-fractional function, under the desired mapping, the point , which is symmetric to a point with respect to the line Re z + Im z = 0 (Fig. 9 ), will go to exactly

ku symmetric to a point with respect to the circle (Fig. 10), which is the image of the line Re z + Im z = 0, under the desired mapping. Consequently, the points go respectively to the points , substituting which into equation (1), we find the desired mapping:

8. Find a conformal mapping of a circle onto a circle that satisfies the conditions , .

Solution. Point 2 is symmetric about the circle point , and the point is symmetric about the circle point -2 . Therefore, under the desired linear-fractional mapping, the points 2 and will go to the points and 2 , respectively. Let some unknown point pass to a point . Then the linear-fractional mapping that takes the points 2, , respectively, to the points , , -2 can be found from the equation

To find, we use the condition and condition , which means that under the desired mapping, the boundary point z = 3 of the circle passes into some boundary point of the circle .

From the first condition

find . Therefore, the complex number –2 has the form

where . From the second condition

we find r = 2. Hence, = 2 + 2 and

When solving applied problems, it often becomes necessary to transform a given area into an area of ​​a simpler form, and in such a way that the angles between the curves are preserved. Transformations endowed with this property make it possible to successfully solve the problems of aero- and hydrodynamics, the theory of elasticity, the theory of fields of various nature, and many others. We restrict ourselves to transformations of flat regions. A continuous map r0 = f(r) of a flat domain into a domain on the plane is said to be conformal at a point if at that point it has the properties of constant expansion and conservation of angles. Open domains are said to be conformally equivalent if there is a one-to-one mapping from one of these domains to the other, conformal at every point. Riemann's theorem. Any two flat open simply connected domains whose boundaries consist of more than one point are conformally equivalent. The main problem in solving specific problems is the construction of an explicit one-to-one conformal mapping of one of them onto the other from given flat regions. One way to solve this problem in the plane case is to use the apparatus of the theory of functions of a complex variable. As noted above, a univalent analytic function with a nonzero derivative performs a conformal mapping of its domain onto its image. When constructing conformal mappings, the following rule is very useful. The principle of boundary matching. Let a single-valued analytic function w = f(z) be given in a simply connected domain R) of the complex plane z, bounded by the contour 7, continuous in the closure 9) and reflecting the contour 7 onto some contour 7" of the complex p/spacity w. If, in this case, the directions bypassing the contour, then the function w - f(z) performs a conformal mapping of the region of the complex plane z onto the region Z1 of the complex plane w bounded by the contour 7" (Fig. 1). The purpose of this section is to use the domains of univalence found earlier for the basic elementary functions of a complex variable to learn how to construct conformal mappings of open singly-connected plane domains that are often encountered in applications, superimposing the upper half-plane and the unit circle (Fig. .2). To make better use of the table below, some simple transformations of the complex plane are useful. Plane transformations that carry out: 1. parallel transfer (shift by a given complex number a) (Fig. 3), Fig.3 2. rotation (by a given angle 3. stretching (fc > 1) or and compression (Fig. 5). Thus, a transformation of the form 0 any circle can be made a unit circle with a center at zero (Fig. 6 ), any half-plane can be made an upper half-plane, any straight line segment can be converted into a segment of the real axis (Fig. 14). cut along the real ray (0, + "> (Plane with cuts along the real rays J -oo, 0] and (I, + oo[ Plane with a cut along the real ray Plane with a cut along the segment (0, 1J No. 21 1 plane with cuts to the rays lying ia of a straight line passing through the origin of coordinates along the real rays ] - "u, 0] and (1. The plane with a cut along the real ray (0, + in (Plane with a cut along the arc of a circle Ixl - 1, lm z\u003e О Plane with a cut along the arc of a circle III - I, Re z > О Plane with a cut along the action to the real ray (0, Plane with a cut along the arc of a circle Plane with a cut along the real ray [C, + co [ No. 25 Half-plane with cuts Half-plane l with a cut along a segment with a cut along an imaginary ray Circle with cuts Circle 1 with a cut along a segment (1/2, 1J #30 Plane with a cut along the segment (-1, 5/4] Circle Izl with cuts along the segments (-1. -1/2] and (1/2, 1] No. 31 Plane with cuts along cuts I -5/4, 5/4] Circle Ijl with symmetrical cuts along the imaginary axis Circle lie with symmetrical cuts along the real axis Exterior of the circle with cuts Appearance unit circle I with a cut along the segment and 11, 2) №34 Plane with a cut along the segment [-1, 5/4] Plane with a cut along the segment I - 5/4, 3/4] w = e "^z The appearance of a single circle Izl > 1 with cuts along segments that are extensions of its diameter Exterior of the unit circle Iwl > 1 with cuts along segments lying on the real axis , cut along the segment (0, i/2) Semicircle, cut along the segment )