How to find intersection and union. Finding the intersection and union of numerical sets

crossing two sets is called the set of all common elements these sets.

Example :
Let's take the numbers 12 and 18. Find their divisors, denoting the entire set of these divisors, respectively, by the letters A and B:
A \u003d (1, 2, 3, 4, 6, 12),
B = (1, 2, 3, 6, 9, 18).

We see that the numbers 12 and 18 have common divisors: 1, 2, 3, 6. Let's denote them with the letter C:
C = (1, 2, 3, 6).

The set C is the intersection of the sets A and B. They write it like this:
A ∩B=C.

If two sets have no common elements, then the intersection of these sets is empty a bunch of.
The empty set is denoted by the sign Ø, and the following notation is used:

X ∩Y = Ø.

Union two sets is the set consisting of all the elements of these sets.

For example, let's return to the numbers 12 and 18 and the set of their elements A and B. First, we write out the elements of the set A, then we add to them those elements of the set B that are not in the set A. We get the set of elements that A and B have in common. Let's denote it with the letter D:

D = (1, 2, 3, 4, 6, 12, 9, 18).

The set D is the union of the sets A and B. It is written like this:

D=A U b.

The main operations performed on sets are addition (Union), multiplication (intersection) and subtraction . These operations, as we will see later, are not identical to operations of the same name performed on numbers.

Definition : Association(or sum) of two sets A and B is a set containing all such and only such elements that are elements of at least one of these sets. The union of sets A and B is denoted as A  B.

This definition means that the addition of sets A and B is the union of all their elements into one set A  B. If the same elements are contained in both sets, then these elements enter the union only once.

The union of three or more sets is defined similarly.

Definition : crossing(or multiplication) of two sets A and B is a set consisting of those and only those elements that belong to the set A and set B at the same time. The intersection of sets A and B is denoted as A  B.

The intersection of three or more sets is defined similarly.

Definition : The difference of sets A and B is the set consisting of those and only those elements of set A that do not belong to set B. The difference of sets A and B is denoted as A \ B. The operation by which the difference of sets is found is called subtraction.

If B  A, then the difference A \ B is called the complement of the set B to the set A. If the set B is a subset of the universal set U, then the complement of B to U is denoted, that is = U\B.

Exercises :

Consider three sets N={0,2,4,5,6,7}, M=(1,3,5,7,9) and P=(1,3,9,11). To find

A= N  M
B=N M
C=N P

Answer which of the operations on the given sets should be used to obtain the sets described below.

Given: BUT- many of all faculty students, AT– many students with academic debts. Define With- a lot of successful students of the faculty.
Given: BUT- a set of all excellent students of the faculty, AT- a lot of students who do not have academic debts, With is the set of successful students with at least one triple. Define D- a lot of students of the faculty who have time without triples.
Given: U is the set of all students of the study group, BUT- a lot of students of this group who received a credit in physical education, AT- many students of the same group who successfully passed the test in the history of the Fatherland. Define With is the set of students from the same study group who excel in both disciplines, D– a set of students of the same group who “failed” at least one of the tests.

Union and intersection properties of sets

From the definitions of union and intersection of sets, the properties of these operations follow, which are presented in the form of equalities that are valid for any sets A , B and With .

A  B = B  A - commutativity of the union;

A  B = B  A - commutativity of the intersection;

A  (B  With ) = (A  B )  With - association association;

A  (B  With ) = (A  B )  With - associativity of the intersection;

A  (B  With ) = (A  B )  (A  WITH) - distributivity of the intersection with respect to the union;

A  (B  With ) = (A  B )  (A  WITH) - distributivity of the union with respect to the intersection;

Absorption laws:

A  A = A

A  A = A

A  Ø = A

A  Ø = Ø

A  U = U

A  U = A

It should be noted that the difference does not have the properties of commutativity and associativity, that is, A \ B ≠ B \ A and A \ (B \ With ) ≠ (A \ B ) \ With . This can be easily verified by constructing the Euler-Venn diagrams.

Sets. Operations on sets.
Set display. Set power

I welcome you to the first lesson in higher algebra, which appeared ... on the eve of the fifth anniversary of the site, after I had already created more than 150 articles in mathematics, and my materials began to take shape in a completed course. However, I will hope that I am not late - after all, many students begin to delve into lectures only for state exams =)

The university course of vyshmat is traditionally based on three pillars:

– mathematical analysis (limits, derivatives etc.)

– and finally the 2015/16 season school year opens with lessons Algebra for dummies, Elements of mathematical logic, on which we will analyze the basics of the section, as well as get acquainted with basic mathematical concepts and common notation. I must say that in other articles I do not abuse "squiggles" , however, this is just a style, and, of course, they need to be recognized in any state =). I inform new readers that my lessons are practice-oriented, and the following material will be presented in this vein. For more complete and academic information, please refer to the textbooks. Go:

A bunch of. Set examples

A set is a fundamental concept not only of mathematics, but of the whole world around. Take any item in your hand right now. Here you have a set consisting of one element.

AT broad sense, a set is a collection of objects (elements) that are understood as a whole(according to certain signs, criteria or circumstances). Moreover, these are not only material objects, but also letters, numbers, theorems, thoughts, emotions, etc.

Sets are usually denoted by large with Latin letters (as an option, with subscripts: etc.), and its elements are written in curly braces, for example:

- a set of letters of the Russian alphabet;
- a bunch of natural numbers;

Well, it's time to get to know each other a little:
– many students in the 1st row

… I am glad to see your serious and focused faces =)

Sets and are final(consisting of a finite number of elements), and a set is an example endless sets. In addition, in theory and practice, the so-called empty set:

is a set that does not contain any element.

The example is well known to you - the set in the exam is often empty =)

The membership of an element in a set is indicated by the symbol , for example:

- the letter "be" belongs to the set of letters of the Russian alphabet;
- the letter "beta" not belongs to the set of letters of the Russian alphabet;
– the number 5 belongs to the set of natural numbers;
- but the number 5.5 is no longer there;
- Voldemar does not sit in the first row (and even more so, does not belong to the set or =)).

In abstract and not so algebra, the elements of a set are denoted by small Latin letters and, accordingly, the fact of belonging is drawn up in the following style:

– the element belongs to the set .

The above sets are written direct transfer elements, but this is not the only way. Many sets are conveniently defined using some sign (s), which is inherent to all its elements. For example:

is the set of all natural numbers less than 100.

Remember: a long vertical stick expresses the verbal turnover "which", "such that". Quite often, a colon is used instead: - let's read the entry more formally: "the set of elements belonging to the set of natural numbers, such that » . Well done!

This set can also be written by direct enumeration:

More examples:
- and if there are quite a lot of students in the 1st row, then such a record is much more convenient than their direct listing.

is the set of numbers belonging to the interval . Note that this refers to the set valid numbers (about them later), which can no longer be listed separated by commas.

It should be noted that the elements of a set do not have to be "homogeneous" or logically related. Take a big bag and start randomly stuffing it into it. various items. There is no regularity in this, but, nevertheless, we are talking about a variety of subjects. Figuratively speaking, a set is a separate “package” in which a certain set of objects turned out to be “by the will of fate”.

Subsets

Almost everything is clear from the name itself: the set is subset set if every element of the set belongs to the set . In other words, a set is contained in a set:

An icon is called an icon inclusion.

Let's return to the example in which is the set of letters of the Russian alphabet. Denote by - the set of its vowels. Then:

It is also possible to single out a subset of consonant letters and, in general, an arbitrary subset consisting of any number of randomly (or non-randomly) taken Cyrillic letters. In particular, any Cyrillic letter is a subset of the set .

Relations between subsets are conveniently depicted using conditional geometric scheme, which is called Euler circles.

Let be a set of students in the 1st row, be a set of group students, and be a set of university students. Then the relation of inclusions can be represented as follows:

The set of students of another university should be depicted as a circle that does not intersect the outer circle; the multitude of the country's students in a circle that contains both of these circles, and so on.

Typical example we observe inclusions when considering numerical sets. Let's repeat the school material, which is important to keep in mind when studying higher mathematics:

Numeric sets

As you know, historically, natural numbers were the first to appear, designed to count material objects (people, chickens, sheep, coins, etc.). This set has already been met in the article, the only thing is that we are now slightly modifying its designation. The fact is that numerical sets are usually denoted by bold, stylized or thickened letters. I prefer to use bold:

Sometimes zero is included in the set of natural numbers.

If we add the same numbers with the opposite sign and zero to the set, we get set of integers:

Rationalizers and lazy people write down its elements with icons "plus minus":))

It is quite clear that the set of natural numbers is a subset of the set of integers:
- since each element of the set belongs to the set . Thus, any natural number can be safely called an integer.

The name of the set is also "talking": integers - this means no fractions.

And, as soon as they are integers, we immediately recall the important signs of their divisibility by 2, 3, 4, 5 and 10, which will be required in practical calculations almost every day:

An integer is divisible by 2 without a remainder if it ends in 0, 2, 4, 6, or 8 (i.e. any even digit). For example, numbers:
400, -1502, -24, 66996, 818 - divided by 2 without a remainder.

And let's immediately analyze the "related" sign: integer is divisible by 4 if the number made up of its last two digits (in their order) is divisible by 4.

400 is divisible by 4 (because 00 (zero) is divisible by 4);
-1502 - not divisible by 4 (because 02 (two) is not divisible by 4);
-24, of course, is divisible by 4;
66996 - divisible by 4 (because 96 is divisible by 4);
818 - not divisible by 4 (because 18 is not divisible by 4).

Make your own simple justification for this fact.

Divisibility by 3 is a little more difficult: an integer is divisible by 3 without a remainder if the sum of its digits is divisible by 3.

Let's check if the number 27901 is divisible by 3. To do this, we sum up its numbers:
2 + 7 + 9 + 0 + 1 = 19 - not divisible by 3
Conclusion: 27901 is not divisible by 3.

Let's sum the digits of the number -825432:
8 + 2 + 5 + 4 + 3 + 2 = 24 - divisible by 3
Conclusion: the number -825432 is divisible by 3

Whole number is divisible by 5, if it ends with a five or a zero:
775, -2390 - divisible by 5

Whole number is divisible by 10 if it ends in zero:
798400 - divisible by 10 (and obviously at 100). Well, probably everyone remembers - in order to divide by 10, you just need to remove one zero: 79840

There are also signs of divisibility by 6, 8, 9, 11, etc., but there is practically no practical sense from them =)

It should be noted that the listed criteria (seemingly so simple) are rigorously proved in number theory. This section of algebra is generally quite interesting, however, its theorems ... just a modern Chinese execution =) And Voldemar at the last desk was enough ... but that's okay, soon we will deal with life-giving exercise =)

The next number set is a bunch of rational numbers :
- that is, any rational number can be represented as a fraction with an integer numerator and natural denominator.

Obviously, the set of integers is subset sets of rational numbers:

Indeed, any integer can be represented as rational fraction, For example: etc. Thus, an integer can quite legitimately be called a rational number.

A characteristic "identifying" sign of a rational number is the fact that when dividing the numerator by the denominator, one gets either
is an integer,

or
– ultimate decimal,

or
- endless periodical decimal (replay may not start immediately).

Admire the division and try to perform this action as little as possible! In the organizational article Higher Mathematics for Dummies and in other lessons I repeatedly repeated, repeat, and will repeat this mantra:

AT higher mathematics we strive to perform all actions in ordinary (correct and improper) fractions

Agree that dealing with a fraction is much more convenient than with decimal number 0,375 (not to mention infinite fractions).

Let's go further. In addition to the rational ones, there are many irrational numbers, each of which can be represented as an infinite non-periodic decimal fraction. In other words, there is no regularity in the "infinite tails" of irrational numbers:
("year of birth of Leo Tolstoy" twice)
etc.

There is plenty of information about the famous constants "pi" and "e", so I do not dwell on them.

The union of rational and irrational numbers forms set of real (real) numbers:

- icon associations sets.

The geometric interpretation of the set is familiar to you - it is a number line:

Each real number corresponds to a certain point of the number line, and vice versa - each point of the number line necessarily corresponds to some real number. Essentially, I have now formulated continuity property real numbers, which, although it seems obvious, is rigorously proved in the course of mathematical analysis.

The number line is also denoted by an infinite interval, and the notation or equivalent notation symbolizes the fact that it belongs to the set of real numbers (or simply "x" - a real number).

With embeddings, everything is transparent: the set of rational numbers is subset sets of real numbers:
, thus, any rational number can be safely called a real number.

The set of irrational numbers is also subset real numbers:

At the same time, subsets and do not intersect- that is, no irrational number can be represented as a rational fraction.

Are there any other number systems? Exist! This, for example, complex numbers, with which I recommend that you read literally in the coming days or even hours.

In the meantime, we turn to the study of set operations, the spirit of which has already materialized at the end of this section:

Actions on sets. Venn diagrams

Venn diagrams (similar to Euler circles) are a schematic representation of actions with sets. Again, I warn you that I will not cover all operations:

1) intersection And and is marked with

The intersection of sets is called a set, each element of which belongs to and set , and set . Roughly speaking, an intersection is a common part of sets:

So, for example, for sets:

If the sets have no identical elements, then their intersection is empty. We just came across such an example when considering numerical sets:

The sets of rational and irrational numbers can be schematically represented by two non-overlapping circles.

The operation of intersection is also applicable to more sets, in particular, Wikipedia has a good an example of the intersection of sets of letters of three alphabets.

2) Union sets is characterized by a logical connection OR and is marked with

A union of sets is a set, each element of which belongs to the set or set :

Let's write the union of sets:
- roughly speaking, here you need to list all the elements of the sets and , and the same elements (in this case, the unit at the intersection of sets) must be specified once.

But the sets, of course, may not intersect, as is the case with rational and irrational numbers:

In this case, you can draw two non-intersecting shaded circles.

The union operation is applicable for more sets, for example, if , then:

The numbers don't have to be in ascending order. (I did this purely for aesthetic reasons). Without further ado, the result can be written like this:

3) difference and does not belong to the set:

The difference is read as follows: “a without be”. And you can argue in exactly the same way: consider the sets. To write down the difference, you need to “throw out” all the elements that are in the set from the set:

Example with numeric sets:
- here all natural numbers are excluded from the set of integers, and the notation itself reads like this: "the set of integers without the set of naturals."

Mirror: difference sets and call the set, each element of which belongs to the set and does not belong to the set:

For the same sets
- from the set "thrown out" what is in the set.

But this difference turns out to be empty: . And in fact - if integers are excluded from the set of natural numbers, then, in fact, nothing will remain :)

In addition, sometimes consider symmetrical the difference that combines both "crescents":
- in other words, it is "everything but the intersection of sets."

4) Cartesian (direct) product sets and is called a set all orderly pairs in which the element and the element

We write the Cartesian product of sets:
- it is convenient to enumerate pairs according to the following algorithm: “first, we sequentially attach each element of the set to the 1st element of the set, then we attach each element of the set to the 2nd element of the set, then we attach each element of the set to the 3rd element of the set»:

Mirror: Cartesian product sets and is called the set of all orderly pairs in which . In our example:
- here the recording scheme is similar: first, we sequentially attach all elements of the set to “minus one”, then to “de” - the same elements:

But this is purely for convenience - in both cases, the pairs can be listed in any order - it is important to write down here all possible couples.

And now the highlight of the program: the Cartesian product is nothing but a set of points in our native Cartesian coordinate system .

Exercise for self-fixing material:

Perform operations if:

A bunch of it is convenient to describe it by listing its elements.

And a fad with intervals of real numbers:

Recall that the square bracket means inclusion numbers into the interval, and round - it exclusion, that is, "minus one" belongs to the set, and "three" not belongs to the set. Try to figure out what the Cartesian product of these sets is. If you have any difficulties, follow the drawing;)

Quick Solution assignments at the end of the lesson.

Set display

Display set to set is rule, according to which each element of the set is associated with an element (or elements) of the set . In the event that it matches the only one element, this rule is called clearly defined function or just function.

The function, as many people know, is most often denoted by a letter - it associates to each element is the only value belonging to the set .

Well, now I will again disturb a lot of students of the 1st row and offer them 6 topics for abstracts (set):

Installed (voluntarily or involuntarily =)) the rule associates each student of the set with a single topic of the abstract of the set.

…and you probably couldn’t even imagine that you would play the role of a function argument =) =)

The elements of the set form domain functions (denoted by ), and the elements of the set - range functions (denoted by ).

The constructed mapping of sets has a very important characteristic: it is one-to-one or bijective(bijection). AT this example it means that to each the student is aligned one unique topic of the essay, and vice versa - for each one and only one student is fixed by the topic of the abstract.

However, one should not think that every mapping is bijective. If the 7th student is added to the 1st row (to the set), then the one-to-one correspondence will disappear - or one of the students will be left without a topic (no display at all), or some topic will go to two students at once. The opposite situation: if a seventh topic is added to the set, then the one-to-one mapping will also be lost - one of the topics will remain unclaimed.

Dear students, on the 1st row, do not be upset - the remaining 20 people after class will go to clean up the territory of the university from autumn foliage. The supply manager will give twenty goliks, after which a one-to-one correspondence will be established between the main part of the group and the brooms ..., and Voldemar will also have time to run to the store =)). unique"y", and vice versa - for any value of "y" we can unambiguously restore "x". Thus, it is a bijective function.

! Just in case, I eliminate a possible misunderstanding: my constant reservation about the scope is not accidental! The function may not be defined for all "x", and, moreover, it may be one-to-one in this case as well. Typical example:

But at quadratic function nothing like this is observed, firstly:
- i.e, various meanings"x" appeared in same meaning "y"; and secondly: if someone calculated the value of the function and told us that , then it is not clear - this “y” was obtained at or at ? Needless to say, there is not even a smell of mutual unambiguity here.

Task 2: view graphs of basic elementary functions and write out bijective functions on a piece of paper. Checklist at the end of this lesson.

Set power

Intuition suggests that the term characterizes the size of the set, namely the number of its elements. And intuition does not deceive us!

The cardinality of the empty set is zero.

The cardinality of the set is six.

The power of the set of letters of the Russian alphabet is thirty-three.

In general, the power of any final set is equal to the number of elements of this set.

... perhaps not everyone fully understands what it is final set - if you start counting the elements of this set, then sooner or later the count will end. What is called, and the Chinese will someday run out.

Of course, sets can be compared in cardinality, and their equality in this sense is called equal power. Equivalence is defined as follows:

Two sets are equivalent if a one-to-one correspondence can be established between them..

The set of students is equivalent to the set of abstract topics, the set of letters of the Russian alphabet is equivalent to any set of 33 elements, etc. Notice exactly what anyone a set of 33 elements - in this case, only their number matters. The letters of the Russian alphabet can be compared not only with many numbers
1, 2, 3, ..., 32, 33, but also in general with a herd of 33 cows.

Things are much more interesting with infinite sets. Infinities are also different! ...green and red The "smallest" infinite sets are counting sets. If it is quite simple, the elements of such a set can be numbered. The reference example is the set of natural numbers . Yes - it is infinite, but each of its elements in PRINCIPLE has a number.

There are a lot of examples. In particular, the set of all even natural numbers is countable. How to prove it? It is necessary to establish its one-to-one correspondence with the set of natural numbers or simply number the elements:

A one-to-one correspondence is established, therefore, the sets are equivalent and the set is countable. It is paradoxical, but from the point of view of power - there are as many even natural numbers as natural ones!

The set of integers is also countable. Its elements can be numbered, for example, like this:

Moreover, the set of rational numbers is also countable. . Since the numerator is an integer (and, as just shown, they can be numbered), and the denominator is a natural number, then sooner or later we will “get” to any rational fraction and assign it a number.

But the set of real numbers is already countless, i.e. its elements cannot be numbered. This fact although obvious, it is rigorously proved in set theory. The cardinality of the set of real numbers is also called continuum, and compared to countable sets, this is a "more infinite" set.

Since there is a one-to-one correspondence between the set and the number line (see above), then the set of points of the real line is also countless. And what's more, there are the same number of points on a kilometer and a millimeter segment! Classic example:

By turning the beam counterclockwise until it coincides with the beam, we will establish a one-to-one correspondence between the points of the blue segments. Thus, there are as many points on the segment as there are on the segment and !

This paradox, apparently, is connected with the mystery of infinity ... but now we will not bother with the problems of the universe, because the next step is

Task 2 One-to-One Functions in Lesson Illustrations

Lesson Objectives:

educational: the formation of skills to identify sets, subsets; the formation of skills to find the area of intersection and union of sets in images and name the elements from this area, solve problems;
developing: development cognitive interest students; development of the intellectual sphere of the individual, the development of skills to compare and generalize.
educational: to cultivate accuracy and attentiveness in making decisions.

During the classes.

1. Organizational moment.

2. The teacher reports the topic of the lesson, together with the students formulates goals and objectives.

3. The teacher, together with the students, recalls the material studied on the topic “Sets” in grade 7, introduces new concepts and definitions, formulas for solving problems.

“Many is many, thought by us as one” (founder of set theory - Georg Cantor). KANTOR (Cantor) Georg (1845-1918) - German mathematician, logician, theologian, creator of the theory of transfinite (infinite) sets, which had a decisive influence on the development of mathematical sciences at the turn of the 19th and 20th centuries.

A set is one of the basic concepts of modern mathematics, used in almost all of its sections.

Unfortunately, the basic concept of the theory - the concept of a set - cannot be given a rigorous definition. Of course, one can say that a set is a "collection", "collection", "ensemble", "collection", "family", "system", "class", etc., however, all this would not be a mathematical definition, but rather the abuse of the vocabulary of the Russian language.

In order to define any concept, it is necessary, first of all, to indicate, as a particular case of which more general concept, it is, it is impossible to do this for the concept of a set, because there is no more general concept than a set in mathematics.

Often you have to talk about several things, united by some sign. So, we can talk about the set of all chairs in the room, about the set of all cells human body, the set of all potatoes in a given bag, the set of all fish in the ocean, the set of all squares on a plane, the set of all points on a given circle, etc.

The objects that make up a given set are called its elements.

For example, the set of days of the week consists of the elements: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday.

Many months - from the elements: January, February, March, April, May, June, July, August, September, October, November, December.

A bunch of arithmetic operations- from elements: addition, subtraction, multiplication, division.

For example, if A means the set of all natural numbers, then 6 belongs to A, but 3 does not belong to A.

If A is the set of all months in a year, then May belongs to A, but Wednesday does not belong to A.

If a set contains a finite number of elements, then it is called finite, and if it has an infinite number of elements, then it is called infinite. So the set of trees in the forest is finite, but the set of points on the circle is infinite.

Paradox in logic- this is a contradiction that has the status of a logically correct conclusion and, at the same time, is a reasoning that leads to mutually exclusive conclusions.

As already mentioned, the concept of a set is at the heart of mathematics. Using the simplest sets and various mathematical constructions, one can construct almost any mathematical object. The idea of building all of mathematics on the basis of set theory was actively promoted by G. Kantor. However, for all its simplicity, the concept of a set is fraught with the danger of contradictions or, as they say, paradoxes. The appearance of paradoxes is due to the fact that not all constructions and not all sets can be considered.

The simplest of paradoxes is " barber's paradox".

One soldier was ordered to shave those and only those soldiers of his platoon who did not shave themselves. Disobeying an order in the army, as you know, is the gravest crime. However, the question arose whether this soldier should shave himself. If he shaves, then he should be attributed to the many soldiers who shave themselves, and he has no right to shave such. If he does not shave himself, then he will fall into the multitude of soldiers who do not shave themselves, and according to the order, he is obliged to shave such soldiers. Paradox.

On sets, as well as on many other mathematical objects, you can perform various operations, which are sometimes called set-theoretic operations or set-operations. As a result of operations, new sets are obtained from the original sets. Sets are denoted by uppercase Latin letters, and their elements by lowercase. Recording a R means that the element a belongs to the set R, i.e a R. Otherwise, when a does not belong to the set R, write a R .

Two sets BUT and AT called equal (BUT =AT) if they consist of the same elements, that is, each element of the set BUT is an element of the set AT and vice versa, each element of the set AT is an element of the set BUT .

Set comparison.

Set A is contained in set B (set B includes set A) if every element of A is an element of B:

They say that many BUT contained in many AT or set BUT is an subset sets AT(in this case write BUT AT) if each element of the set BUT is also an element of the set AT. This relationship between sets is called inclusion . For any set BUT there are inclusions: Ø BUT and BUT BUT

In this case A called subset B, B - superset A. If , then A called own subset AT. notice, that ,

A-priory ,

The two sets are called equal if they are subsets of each other

Operations on sets

intersection.

Union.

Properties.

1. The operation of union of sets is commutative

2. The operation of union of sets is transitive

3. The empty set X is a neutral element of the operation of union of sets

1. Let A = (1,2,3,4),B = (3,4,5,6,7). Then

2. A \u003d (2,4,6,8,10), B \u003d (3,6,9,12). Let's find the union and intersection of these sets:

{2,4,6,8, 10,3,6,9,12}, = {6}.

3. The set of children is a subset of the total population

4. The intersection of the set of integers with the set of positive numbers is the set of natural numbers.

5. The union of the set of rational numbers with the set of irrational numbers is the set of positive numbers.

6. Zero is the complement of the set of natural numbers with respect to the set of non-negative integers.

Venn diagrams(Venn diagrams) - common name a number of visualization methods and methods of graphic illustration, widely used in various fields of science and mathematics: set theory, in fact "venn diagram" shows all possible relationship between sets or events from some family; varieties venn diagrams are: Euler diagrams,

Venn diagram of four sets.

Actually "venn diagram" shows all possible relationships between sets or events from some family. The usual Venn diagram has three sets. Venn himself tried to find graceful way with symmetrical shapes representing on the diagram more sets, but he was only able to do this for four sets (see figure on the right) using ellipses.

Euler diagrams

Euler diagrams are similar to Venn diagrams. Euler diagrams can be used to evaluate the likelihood of set-theoretic identities.

Task 1. There are 30 people in the class, each of whom sings or dances. It is known that 17 people sing, and 19 people know how to dance. How many people are singing and dancing at the same time?

Decision: First, we note that out of 30 people, 30 - 17 = 13 people cannot sing.

They all know how to dance, because according to the condition, each student of the class sings or dances. In total, 19 people can dance, 13 of them cannot sing, which means that 19-13 = 6 people can dance and sing at the same time.

Problems on the intersection and union of sets.

Sets A = (3.5, 0, 11, 12, 19), B = (2.4, 8, 12, 18.0) are given.
Find the sets AU B,
Make up at least seven words whose letters form subsets of the set
A - (k, a, p, y, s, e, l, b).
Let A be the set of natural numbers divisible by 2, and B be the set of natural numbers divisible by 4. What conclusion can be drawn about these sets?
The company employs 67 people. Of these, 47 know English language, 35 are German and 23 are both languages. How many people in the company do not speak English or German?
Of the 40 students in our class, 32 like milk, 21 like lemonade, and 15 like both milk and lemonade. How many kids in our class don't like milk or lemonade?
12 of my classmates like to read detective stories, 18 love to read science fiction, three of them read both with pleasure, and one does not read anything at all. How many students are in our class?
Of those 18 of my classmates who like to watch thrillers, only 12 are not averse to watching cartoons as well. How many of my classmates watch only "cartoons" if there are 25 students in our class, each of whom likes to watch either thrillers, or cartoons, or both?
Of the 29 boys in our yard, only two do not go in for sports, and the rest attend football or tennis sections, or even both. There are 17 boys playing football and 19 playing tennis. How many football players play tennis? How many tennis players play football?
65% of Grandma's Rabbits love carrots, 10% love both carrots and cabbage. How many percent of rabbits are not averse to eating cabbage?
There are 25 students in one class. Of these, 7 love pears, 11 love cherries. Two like pears and cherries; 6 - pears and apples; 5 - apples and cherries. But there are two students in the class who love everything and four who don't like fruit at all. How many students in this class like apples?
22 girls participated in the beauty contest. Of these, 10 were beautiful, 12 were smart and 9 were kind. Only 2 girls were both beautiful and smart; 6 girls were smart and kind at the same time. Determine how many beautiful and at the same time kind girls were, if I tell you that among the participants there was not a single smart, kind and at the same time beautiful girl?
There are 35 students in our class. For the first quarter of the five in the Russian language, 14 students had; in mathematics - 12; in history - 23; in Russian and mathematics - 4; in mathematics and history - 9; in Russian language and history - 5. How many students have fives in all three subjects, if there is not a single student in the class who does not have fives in at least one of these subjects?
Out of 100 people, 85 speak English, 80 speak Spanish, and 75 speak German. All speak at least one foreign language. Among them there are no those who know two foreign languages, but there are those who speak three languages. How many of those 100 people know three languages?
Of the company's employees, 16 visited France, 10 - Italy, 6 - England; in England and Italy - 5; in England and France - 6; in all three countries - 5 employees. How many people have visited both Italy and France, if there are 19 people in the company, and each of them has visited at least one of these countries?