Limited number set

The set of real numbers is called bounded above if there is a number such that all elements do not exceed:

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See what “Limited set” is in other dictionaries: 1) O. m. in a metric space X (with a metric) is a set A whose diameter is finite. 2) O. m. in topological. vector space E(over a field k) is a set B such that it is absorbed by every neighborhood of zero U (i.e., there exists such a one). M.I.... ...

Mathematical Encyclopedia E(over a field k) is a set B such that it is absorbed by every neighborhood of zero U (i.e., there exists such a one). M.I.... ...

In a metric space it is the same as a completely bounded subspace of a given metric. space. See quite limited space. A. V. Arkhangelsky ... IN mathematical analysis

, and related branches of mathematics, a bounded set is a set that in a certain sense has a finite size. The basic concept is the boundedness of a numerical set, which is generalized to the case... ... Wikipedia a bunch of - set set - set One of the basic concepts of modern mathematics, “an arbitrary collection of defined and distinguishable objects, mentally united into a single... ...

Technical Translator's Guide A bunch of - one of the basic concepts of modern mathematics, “an arbitrary collection of defined and distinguishable objects, mentally united into a single whole.” (This is how set was defined by the founder of set theory, the famous German... ...

Economic and mathematical dictionary See Class in Logic. Philosophical encyclopedic Dictionary . M.: Soviet encyclopedia . Ch. editor: L. F. Ilyichev, P. N. Fedoseev, S. M. Kovalev, V. G. Panov. 1983. MANY...

Philosophical Encyclopedia

This term has other meanings, see Set (meanings). A set, a type and data structure in computer science, is an implementation of the mathematical object set. Data type set allows you to store a limited number of values... ... Wikipedia 1) P. m. analytical function E(over a field k) is a set B such that it is absorbed by every neighborhood of zero U (i.e., there exists such a one). M.I.... ...

f(z) of complex variables z=(z1,...,zn), n 1, is a set of Ppoints of a certain domain of D complex space C n such that: a) f(z) is holomorphic everywhere in; b) f(z) does not continue analytically to any point P; c) for... ...- the object of the study itself is not the sublanguage itself, but a certain set of texts, which is in principle infinite or, in any case, open. It is set descriptively, by characterizing the sources of these texts. They are the ones... ... Explanatory translation dictionary

Let's consider the arrangement of the graphs relative to each other inverse functions V Cartesian system coordinates and prove the following statement.

Lemma 1.1. If a, b R, then the points M 1 (a, b), M 2 (b, a) of the plane are symmetrical with respect to the straight line y = x.

If a = b, then the points M1, M2 coincide and lie on the straight line y = x. We will assume that a 6= b. The line passing through the points M1, M2 has the equation y = −x+a+b, and therefore is perpendicular to the line y = x.

Since the middle of the segment M1 M2 has coordinates a + 2 b ,a + 2 b ! , That

it lies on the straight line y = x. Therefore, points M1, M2

Consequence. If the functions f: X −→ Y and ϕ : Y −→ X are mutually inverse, then their graphs are symmetrical with respect to the straight line y = x if they are plotted in the same coordinate system.

Let f = ((x, f(x)) | x X),ϕ = ((y, ϕ(y)) | y Y ) be the graphs of the functions f and ϕ, respectively. Because

(a, b) f (b = f(a), a X) (a = ϕ(b), b Y) (b, a)ϕ ,

then, by virtue of the proven lemma, the graphs f and ϕ symmetrical about the straight line y = x.

1.6 Properties of numerical sets

1.6.1 Bounded number sets

Definition 1.26. Let X be non-empty number set. A set X is said to be bounded above (below) if there is a number a such that x 6 a (x > a ) for any element x X . In this case, the number a is called the upper (lower) boundary of the set X. A set bounded below and above is called bounded.

Using logical symbols, the upper boundedness of a set X is written as follows:

a R: x 6 a, x X.

Taking into account the properties of the modulus of a number, we can give the following equivalent definition of a bounded set.

Definition 1.27. A non-empty number set X is called bounded if there is a positive number M such that

Definition 1.28. An element from a numerical set X is called a maximum (minimum) element in X if x 6 a (respectively, x > a) for any x of X, and they write: a = max X (respectively, a = min X).

By virtue of the order axiom (3.b), it is easy to show that if a set X in R has a maximal (minimal) element, then it is unique.

Note that if a number set X has a maximum (minimal) element a, then it is bounded above (below) and the number a is the upper (lower) boundary of the set X. However, not every number set bounded above (below) has a maximum (minimal) element .

Example 1.5. Let us show that the set X = = inf (a, b) = a.

These examples show, in particular, that the lower and upper faces may or may not belong to the set itself.

By their very definition, the upper and lower bounds of a set are unique. In fact, if in some set, even belonging to the extended number line, there is a smallest (largest) element, then it is unique, since of two different elements of the set, the larger of them cannot be the smallest element, and the smaller cannot be the largest.

Does a set bounded above (below) always have an exact upper (lower) bound? Indeed, since there are infinitely many upper (lower) bounds, and among the infinite set of numbers there is not always the largest (smallest), the existence of the supremum (infinum) requires special proof.

Theorem 7.3(1)

Every non-empty set bounded above has an upper bound, and every non-empty set bounded below has a lower bound.

Proof

Let a non-empty number set A be bounded above, B be the set of all numbers bounding set A from above. If then from the definition of a number bounding above

set, it follows that a≤b. Therefore, by the property of continuity real numbers there is a number β such that the inequality a≤β≤b holds for all. Inequality means that the number β bounds the set A from above, and the inequality means that the number β is the smallest among all the numbers that bound the set A from above. Therefore, β = sup A.

It is proved in a similar way that a numerical set bounded from below has an infimum.

Sets whose elements are points are called point sets. Thus, we can talk about point sets on a line, on a plane, in any space. For the sake of simplicity, we will limit ourselves to considering point sets on a line.

Between real numbers and points on a line there is close connection: each real number can be assigned a point on a line and vice versa. Therefore, when talking about point sets, we will also include among them sets consisting of real numbers - sets on the number line. Conversely: in order to define a point set on a line, we will usually specify the coordinates of all points of our set.

Point sets (and, in particular, point sets on a line) have a number of special properties, distinguishing them from arbitrary sets and distinguishing the theory of point sets into an independent mathematical discipline. First of all, it makes sense to talk about the distance between two points. Further, order relationships can be established between points on a straight line (to the left, to the right); in accordance with this, they say that a point set on a line is an ordered set. Finally, as noted above, Cantor’s principle is valid for the straight line; This property of a line is usually characterized as the completeness of a line.

Let us introduce notation for the simplest sets on a line.

A segment is a set of points whose coordinates satisfy the inequalities.

An interval is a set of points whose coordinates satisfy the conditions.

Half-intervals and are determined accordingly by the conditions: and .

Intervals and half-intervals can be improper. Namely, it denotes the entire line, and, for example, the set of all points for which .

Let's start by considering the various possibilities for arranging the set as a whole on a straight line.

Bounded and Unbounded Sets

The set of points on a line can either consist of points whose distances from the origin do not exceed a certain positive number, or have points arbitrarily far from the origin. In the first case, the set is called bounded, and in the second - unbounded. An example of a bounded set is the set of all points on a segment, and an example of an unbounded set is the set of all points with integer coordinates.

It is easy to see that if is a fixed point on a line, then the set will be bounded if and only if the distances from the point to any point do not exceed some positive number.

Sets bounded above and below

Let be a set of points on a line. If there is a point on a line such that any point is located to the left of the point, then they say that the set limited from above. Similarly, if there is a point on a line such that any point is located to the right of the point, then the set is called bounded below. Thus, the set of all points on a line with positive coordinates is bounded below, and the set of all points with negative coordinates is bounded above.

It is clear that the above definition of a bounded set is equivalent to the following: a set of points on a line is called bounded if it is bounded above and below. Despite the fact that these two definitions are very similar to each other, there is a significant difference between them: the first is based on the fact that a distance is defined between points on a straight line, and the second that these points; form an ordered set.

We can also say that a set is limited if it is entirely located on a certain segment.

Upper and lower bounds of a set

Let the set be bounded above. Then there are points on the line to the right of which there is no point in the set. Using Cantor's principle, we can show that among all points having this property, there is the leftmost one. This point is called the upper bound of the set. The infimum of a point set is defined similarly.

If there is a rightmost point in the set, then it will obviously be the upper bound of the set. However, it may happen that there is no rightmost point in the set. For example, a set of points with coordinates

is bounded from above and does not have a rightmost point. In this case top edge does not belong to the set, but there are points of the set arbitrarily close to. In the above example.

Location of a point set near any point on a line

Let be a point set and be some point on a line. Let's consider various possibilities for locating the set near the point. The following cases are possible:

1. Neither the point nor the points close enough to it belong to the set.

2. The point does not belong to , but there are points of the set arbitrarily close to it.

3. A point belongs to , but all points close enough to it do not belong to .

4. The point belongs to , and there are other points of the set arbitrarily close to it.

In case 1, the point is called external to the set, in case 3 - an isolated point of the set, and in cases 2 and 4 - a limit point of the set.

Thus, if , then the point can be either external to or limiting for it, and if , then it can be either an isolated point of the set or its limit point.

A limit point may or may not belong to the set and is characterized by the condition that there are points of the set arbitrarily close to it. In other words, a point is a limit point of a set if any interval containing the point contains infinitely many points of the set. The concept of a limit point is one of the very important concepts in the theory of point sets.

If a point and all points sufficiently close to it belong to the set, then such a point is called interior point of the set. Any point that is neither external nor internal is called boundary point of the set .

Let's give a few examples to explain all these concepts.

Example 1. Let the set consist of points with coordinates

Then each point of this set is its isolated point, point 0 is a limit point (not belonging to this set), and all other points on the line are external to .

Example 2. Let the set consist of all rational points segment This set has no isolated points, each point on the segment is a limit point, and all other points on the line are external to. It is clear that among the limit points of a set there are both those that belong to it and those that do not belong to it.

Example 3. Let the set consist of all points of the segment. As in the previous example, the set does not have isolated points, and each point on the segment is its limit point. However, unlike the previous example, all limit points belong to this set.

Example 4. Let the set consist of all points with integer coordinates on a line. Each point is its isolated point; the set has no limit points.

Note also that in example 3, every point of the interval is an interior point, and in example 2, every point of the segment is a boundary point.

From the above examples it is clear that infinite set points on a line can have isolated points, or may not have them; in the same way it can have interior points and may not have them. As for limit points, only the set of example 4 does not have a single limit point. As the next important theorem shows, this is due to the fact that the set is unbounded.

Bolzano-Weierstrass theorem

Every bounded infinite set of points on a line has at least one limit point.

Let's prove this theorem. Let be a bounded infinite set of points on a line. Since the set is limited, it is entirely located on a certain segment. Let's divide this segment in half. Since the set is infinite, at least one of the resulting segments contains infinitely many points of the set. Let us denote this segment by (if both halves of the segment contain infinitely many points of the set , then we can denote, for example, the left one by). Next, divide the segment into two equal segments. Since the part of the set located on the segment is infinite, at least one of the resulting segments contains infinitely many points of the set. Let us denote this segment by . Let us continue the process of dividing segments in half indefinitely and each time we will take the half that contains infinitely many points of the set. We will get a sequence of segments. This sequence of segments has the following properties: each subsequent segment is contained in the previous one; each segment contains infinitely many points of the set; the lengths of the segments tend to zero. The first two properties of the sequence directly follow from its construction, and to prove the last property it is enough to note that if the length of the segment is equal to , then the length of the segment is equal to . By virtue of Cantor's principle, there is a single point that belongs to all segments. Let us show that this point is the limit point of the set. To do this, it is enough to establish that if there is some interval containing a point, then it contains infinitely many points of the set. Since each segment contains a point and the lengths of the segments tend to zero, then for sufficiently large segment will be entirely contained in the interval . But by condition it contains infinitely many points of the set. Therefore, it contains infinitely many points of the set. So, the point is indeed a limit point of the set, and the Bolzano-Weierstrass theorem is proven.