Set mappings. Functions Mapping sets general concepts functions basic definitions

Correspondence between sets A and B is a subset of their Cartesian product

In other words, pairs define a correspondence between sets A=( ​​) and B=( ), if rule R is specified, according to which an element from set B is selected for an element of set A.

If an element is associated with some element, b is called way element a and is written as follows: b = R (a). Then - prototype element, which has the properties of uniqueness and completeness:

1. Each prototype corresponds to a single image;

2. The image must be complete, just as the prototype must be complete.

Example. If A is a set of parabolas, B is a set of points on a plane, and R is the correspondence “vertex of a parabola,” then R (a) is a point that is the vertex of parabola a, and consists of all parabolas with a vertex at point b (Fig. 6)

The image of the set A with correspondence R is called set of meanings This correspondence is denoted by R (A) if R (A) consists of the images of all elements of the set A.

The inverse image of the set B with some correspondence R is called domain of definition this correspondence is denoted by . In turn is reverse matching for R.

Thus, for the correspondence of R, specified by the points of the coordinate plane, the domain of definition is the set of points of the abscissa axis, and the set of values ​​is the projection of the points onto the ordinate axis (Fig. 7). Therefore, for some point

M (x, y) y is an image, and x is an inverse image for some correspondence R: Y = R (x), Correspondence between sets X is conveniently in the form of a point on a plane using the Cartesian coordinate method.

Let the correspondence between R and Y=R (X) be given. It corresponds to points M with coordinates (x; y) (Fig. 7). Then the set of points of the plane distinguished by the mapping R will be schedule.

To describe correspondences between sets, the concept of mapping (function) of one set to another is used.

To set the display you must specify:

1. The set that is mapped (the domain of definition of a given map, often denoted by );

2. The set in (on) which a given domain of definition is mapped (the set of values ​​of this mapping is often denoted by );

3. The law or correspondence between these sets, according to which elements (images) from the second set are selected for the elements of the first set (prototypes, arguments).

Designations: .

Methods for specifying displays: analytical(in the form of formulas), tabular, graphic(diagrams or graphs).

There are two main types of single-valued mappings (functions). By power they are divided into surjective And injective.

1. A correspondence in which each element of set A is indicated by a single element of set B, and each element of set B can be indicated at least one element of set A, is called a mapping of set A to set B(surjection).

2. A correspondence in which each element of set A corresponds to a single element of set B, and each element of B corresponds to at most one preimage from A, is called a mapping of set A in many B (injection).

A mapping from set A to set B, in which each element of set B corresponds to a single element of set A, is called one-to-one correspondence between two sets, or bijection.injection and surjection.

Elements of set theory

Concept of set

In mathematics there are a wide variety of sets. We can talk about the set of faces of a polyhedron, points on a line, the set of natural numbers, etc. The concept of set is one of the primary concepts that are not defined through other, simpler ones. Instead of the word “set”, they sometimes say “collection”, “collection” of objects, etc. The objects that make up a given set are called elements of the given set.

Set theory is mainly devoted to the study of infinite sets. Theory finite sets sometimes called combinatorics.

But the simplest properties of sets, those that we will only talk about here, in most cases apply equally to both finite and infinite sets.

Note that in mathematics a set is allowed for consideration that does not contain elements - the empty set. Record AÎ X means that A is an element of the set X.

Definition. The set B is called subset set A if every element of set B is at the same time an element of set A.

Each individual element of set A forms a subset consisting of that one element. Moreover, the empty set is a subset of every set.

A subset of set A is called not your own, if it coincides with set A.

If the set B is a subset of the set A, then we say that B is contained in A and denote B Í A. The subset B of the set A is called own a subset if B is not empty and does not coincide with A (that is, there is an element of the set A that is not contained in B).

Set Operations

Let A and B be arbitrary sets.

Definition. The union of two sets A and B is a set C = AÈB, consisting of all elements belonging to at least one of the sets A and B (see Fig. 1).

The union of any (finite or infinite) number of sets is defined similarly: if A i are arbitrary sets, then their union is a collection of elements, each of which belongs to at least one of the sets A i.

Fig.1 Fig.2

Definition. The intersection of the sets A and B is the set C = AÇB, consisting of all elements belonging to both A and B (see Fig. 2). The intersection of any (finite or infinite) number of sets A i is the set of elements belonging to each of the sets A i.

The operations of union and intersection of sets are by definition commutative and associative, i.e.

AÈB = B È A, (A ÈB) ÈC = A È (B È C),

A Ç B = B Ç A, (A Ç B) Ç C = A Ç (B Ç C).

In addition, they are mutually distributive:

(A È B) Ç C = (A Ç C) È (B Ç C), (1)

(A Ç B) È C = (A È C) Ç (B È C). (2)

Definition. By difference sets A and B is the set of those elements from A that are not contained in B ( rice. 3).

The concept of function. Displaying Sets

Let X and Y be two arbitrary sets.

Definition. They say that a function is defined on X f, taking a value from Y if each element xÎ X is associated with one and only one element yО Y. In this case, the set X is called domain of definition given function, and the set Y is its range of values.

For sets of arbitrary nature, instead of the term “function”, the term “mapping” is often used, speaking about the mapping of one set to another.

If A element from X, then the corresponding element b = f(A) from Y is called way a when displayed f. The totality of all those elements A of X, the image of which is the given element bО Y, called prototype(or more precisely a complete prototype) element b and is designated f –1 (b).

Let A be some set from X; set ( f (A): AÎ A) all elements of the form f (A), Where AÎ A, is called the image of A and is denoted f(A). In turn, for each set B from Y its complete inverse image is determined f–1 (V), namely: f–1 (B) is the collection of all those elements from X whose images belong to B.

Definition. Let's say that f is a mapping from set X to set Y if f(X) = Y; such a mapping is called surjection. In the general case, i.e. When f(X) М Y, they say that f there is a mapping in Y. If for any two distinct elements X 1 and X 2 of X their images y 1 = f (x 1) and y 2 = f (x 2) are also different, then f called injection. Display f: X®Y, which is both a surjection and an injection, is called one-to-one correspondence between X and Y.

Let's consider another important special case of the general concept of correspondence - mapping of sets. If compliant R between sets X And Y element image AX may be empty, or may contain several elements.

Relationship between elements of sets X And Y called display X VY , if each element X from many X only one element of the set matches Y. This element is called element imageX with this display: f(x). On a graph of such a mapping from each point of the set X Only one arrow will come out (Fig. 29).

Consider the following example . Let X- many students in the audience, and Y- many chairs in the same auditorium. Match "student" X sitting on a chair at» sets display X VY. Student image X is a chair.

Let X = Y = N- a set of natural numbers. Matching "decimal notation of a number" X comprises at digits" determines the display N V N. With this display, the number 39 corresponds to the number 2, and the number 45981 corresponds to the number 5 (39 is a two-digit number, 45981 is a five-digit number).

Let X- many quadrilaterals, Y- many circles. Matching "quadrangle" X inscribed in a circle at» is not a display X V Y, since there are quadrilaterals that cannot be inscribed in a circle. But in this case they say that the result is a mapping from the set X into the multitude Y.

If display X V Y such that each element y from many
Y matches one or more elements X from many X, then such a mapping is called display of the set X for manyY.

A bunch of X is called the domain of definition of the mapping f: XY, and a lot Y- the arrival region of this mapping. Part of the arrival area consisting of all images y from many Y, called the mapping value set f.

If y=f(x), then x is called prototype of element y when displayed f. The set of all preimages of an element at they call it a complete prototype: f(y).

Displays are of the following types: injective, surjective and bijective.

If the complete prototype of each element yY contains at most one element (may be empty), then such mappings are called injective.

Displays XY such that f(X)=Y, are called mappings X for the whole multitude Y or surjective(from each point of the set X an arrow comes out, and after changing direction at each point of the set X ends) (Fig. 31).

If a mapping is injective and surjective, then it is called one-to-one or bijective.

Set display X is called a set bijective, if each element XX matches a single element yY, and each element yY matches only one element XX(Fig. 32) .

Bijective mappings generate equal sets : X~Y.

Example . Let - X many coats in the wardrobe, Y- a lot of hooks there. Let's match each coat with the hook on which it hangs. This correspondence is a mapping X inY. It is injective if no hook has more than one coat hanging on it or some hooks are free. This mapping is surjective if all the hooks are occupied or some have several coats hanging on them. It will be bijective if there is only one coat hanging on each hook.

Surjection, injection and bijection

The rule defining the mapping f: X (or the function /) can be conventionally represented by arrows (Fig. 2.1). If there is at least one element in the set Y that none of the arrows points to, then this indicates that the range of values ​​of the function f does not fill the entire set Y, i.e. f(X) C Y.

If the range of values ​​/ coincides with Y, i.e. f(X) = Y, then such a function is called surjective) or, in short, surjection, and the function / is said to map the set X onto the set Y (in contrast to the general case of mapping the set X into the set Y according to Definition 2.1). So, / : X is a surjection if Vy 6 Y 3x € X: /(x) = y. In this case, in the figure, at least one arrow leads to each element of the set Y (Fig. 2.2). In this case, several arrows may lead to some elements from Y. If no more than one arrow leads to any element y € Y, then / is called an injective function, or injection. This function is not necessarily surjective, i.e. the arrows do not lead to all elements of the set Y (Fig. 2.3).

• So, the function /: X -Y Y is an injection if any two different elements from X have as their images when mapping / two different elements from Y, or Vy £ f(X) C Y 3xeX: f(x) = y. Surjection, injection and bijection. Reverse mapping. Composition of mappings is a product of sets. Display schedule. The mapping /: X->Y is called bijective, or bi-jection, if each element of y 6 Y is the image of some and the only element from X, i.e. Vy € f(X) = Y E!x € X: f(x) = y.

In fact, the function / in this case establishes a one-to-one correspondence between the sets X and Y, and therefore it is often called a one-to-one function. Obviously, a function / is bijective if and only if it is both injective and surjective. In this case, arrows (Fig. 2.4) connect in pairs each element from X with each element from Y. Moreover, no two elements from X can be connected by an arrow to the same element from Y, since / is injective, and no two elements from Y cannot be connected by arrows to the same element from X due to the uniqueness requirement of the image in Definition 2.1 of the mapping. Each element of X participates in a pairwise connection, since X is the domain of the function /. Finally, each element from Y also participates in one of the pairs, because / is surjective. The roles of X and Y in this case seem to be completely identical, and if we turn all the arrows back (Fig. 2.5), we get a different mapping or a different function d), which is also injective and surjective. Mappings (functions) that allow such inversion will play an important role in what follows.

In a particular case, the sets X and Y may coincide (X = Y). Then the bijective function will map the set X onto itself. The bijection of a set onto itself is also called a transformation. 2.3. Inverse mapping Let /: X -? Y is a certain bijection and let y € Y. Let us denote by /_1(y) the only element x € X such that /(r) = y. Thus we define some mapping 9: Y Xу which is again a bijection. It is called the inverse mapping, or inverse bijection to /. Often it is also simply called the inverse function and is denoted /"*. In Fig. 2.5, the function d is precisely the inverse of /, i.e. d = f"1.

Examples of solutions in problems

The mappings (functions) / and are mutually inverse. It is clear that if a function is not a bijection, then its inverse function does not exist. Indeed, if / is not injective, then some element y € Y can correspond to several elements x from the set X, which contradicts the definition of a function. If / is not surjective, then there are elements in Y for which there are no preimages in X, i.e. for these elements the inverse function is not defined. Example 2.1. A. Let X = Y = R - a set of real numbers. The function /, defined by the formula y = For - 2, i,y € R, is a bijection. The inverse function is x = (y + 2)/3. b. The real function f(x) = x2 of a real variable x is not surjective, since negative numbers from Y = R are not images of elements from X = K as /: Γ -> Y. Example 2.2. Let A" = R, and Y = R+ be the set of positive real numbers. The function f(x) = ax, a > 0, af 1, is a bijection. The inverse function will be Z"1 (Y) = 1°8a Y

• Surjection, injection and bijection. Reverse mapping. Composition of mappings is a product of sets. Display schedule. 2.4. Composition of mappings If f:X-*Y and g:Y-*Zy then the mapping (p:X -+Z, defined for each a: 6 A" by the formula =, is called a composition (superposition) of mappings (functions) / and d> or a complex function, and is designated rho/ (Fig. 2.6).
• Thus, a complex function before f implements the rule: i Apply / first, and then di, i.e. in the composition of operations “before / you must start with the operation / located on the right. Note that the composition Fig. 2.6 mappings are associative, i.e. if /: X -+Y, d: Y Z and h: Z-*H> then (hog)of = = ho(gof)i which is easier to write in the form ho to /. Let's check this as follows: On any wK "oaicecmee X there is defined a mapping 1x -X X, called identical, often also denoted by idx and given by the formula Ix(x) = x Vx € A". Its -action is that it leaves everything to in their places.

Thus, if is a bijection inverse to the bijection /: X - + Y, then /"1o/ = /x, and /o/-1 = /y, where and /y are identical maps of the sets X and Y, respectively. Conversely, if the mappings f: X ->Y and p: Y A" are such that gof = Ix and fog = /y, then the function / is a bijection, and y is its inverse bijection. Obviously, if / is a bijection of A" onto Y, and $ is a bijection of Y onto Z, then gof is a bijection of X onto Z, and will be the inverse bijection with respect to it. 2.5. Product of sets. Mapping graph Recall that two mutually perpendicular coordinate axes with a scale that is the same for both axes define a rectangular Cartesian coordinate system on the plane (Fig. 2.7).The point O of the intersection of the coordinate axes is called the origin* of coordinates.

Each point M can be associated with a pair (i, y) of real numbers where x is the coordinate of the point Mx on the coordinate axis Ox, and y is the coordinate of the point Mu on the coordinate axis Oy. Points Mx and Mu are the bases of perpendiculars dropped from point M on the Ox and Oy axes, respectively. The numbers x and y are called the coordinates of point M (in the selected coordinate system), and x is called the abscissa of point M, and y is the ordinate of this point. It is obvious that each pair (a, b) of real numbers a, 6 6R corresponds to a point M on the plane, which has these numbers as its coordinates. And conversely, each point M of the plane corresponds to a pair (a, 6) of real numbers a and 6. In the general case, pairs (a, b) and (6, a) define different points, i.e. It is important which of the two numbers a and b comes first in the designation of the pair. Thus, we are talking about an ordered pair. In this regard, the pairs (a, 6) and (6, a) are considered equal to each other, and they define the same point on the plane, if only a = 6. Surjection, injection and bijection. Reverse mapping.

Composition of mappings is a product of sets. Display schedule. The set of all pairs of real numbers, as well as the set of points in the plane, is denoted by R2. This designation is associated with the important concept in set theory of a direct (or dek-artov) product of sets (often they simply speak of a product of sets). Definition 2.2. The product of the sets A and B is the set Ax B of possible ordered pairs (x, y), where the first element is taken from A and the second from B, so that the equality of two pairs (x, y) and (&", y") is determined conditions x = x" and y = y7. Pairs (i, y) and (y, x) are considered different if xy. This is especially important to keep in mind when the sets A and B coincide. Therefore, in the general case A x B f B x A, i.e. the product of arbitrary sets is not commutative, but it is distributive with respect to the union, intersection and difference of sets: where denotes one of the three named operations. The product of sets differs significantly from the indicated operations on two sets. The result of performing these operations is a set whose elements (if it is not empty) belong to one or both of the original sets. The elements of the product of sets belong to the new set and represent objects of a different kind compared to the elements of the original sets. Similar to Definition 2.2

We can introduce the concept of a product of more than two sets. The sets (A x B) x C and A*x (B x C) are identified and simply denoted A x B x C, so. Works Ah Au Ah Ah Ah Ah, etc. denoted, as a rule, by A2, A3, etc. Obviously, the plane R2 can be considered as the product R x R of two copies of the set of real numbers (hence the designation of the set of points of the plane as the product of two sets of points on the number line). The set of points in geometric (three-dimensional) space corresponds to the product R x R x R of three copies of the set of points on the number line, denoted R3.

• The product of n sets of real numbers is denoted by Rn. This set represents all possible collections (xj, X2, xn) of n real numbers X2) xn £ R, and any point x* from Rn is such a collection (xj, x, x*) of real numbers xn £ K*
• The product of n arbitrary sets is a set of ordered collections of n (generally heterogeneous) elements. For such sets, the names tuple or n-ka are used (pronounced “enka”). Example 2.3. Let A = (1, 2) and B = (1, 2). Then the set A x B can be identified with four points of the plane R2, the coordinates of which are indicated when listing the elements of this set. If C = ( 1,2) and D = (3,4), then Example 2.4 Let Then The geometric interpretation of the sets E x F and F x E is presented in Fig. 2.8 . # For the mapping /: X, we can create a set of ordered pairs (r, y), which is a subset of the direct product X x Y.
• Such a set is called the graph of the mapping f (or the graph of the function i*" - Example 2.5. In the case of XCR and Y = K, each ordered pair specifies the coordinates of a point on the plane R2. If X is an interval of the number line R, then the graph of the function can represent some line (Fig. 2.9) Example 2.6 It is clear that with XCR2 and Y = R the graph of the function is a certain set of points in R3, which can represent a certain surface (Fig. 2.10).

If X C R, and Y = R2, then the graph of the function is also a set of points in R3, which can represent a certain line intersected by the plane x = const at only one point M with three coordinates x) yi, y2 (Fig. 2.11) . # All mentioned examples of function graphs are the most important objects of mathematical analysis, and in the future they will be discussed in detail.