Write down the sets that are shaded. Sets

Goals and objectives of the lesson:

Educational:

• repeat and consolidate the ideas received:
• about a set, an element of a set, a subset, the intersection of sets, the union of sets;
• consolidate skills:
• determine the membership of elements in a set and its subset(s), as well as in a set that is the intersection or union of sets;
• find on the diagram the area of ​​elements that do not belong to the set, as well as the area of ​​the set that is the intersection or union of sets and name the elements from this area;
• determine the nature of the relationship between two given sets (set-subset, have intersection, do not have intersection);
• correctly depict the proposed situation;
• computer skills in the graphic editor Paint.

Educational:

• promote the development in children of the ability to observe, compare, and generalize;
• teach children to reason and prove;
• promote the development of thinking, memory, attention;
• promote speech development;
• develop students’ cognitive activity;
• develop interest in the subject;
• develop skills to work on a personal computer.

Educators:

• cultivate friendly relations in the student body;
• cultivate cognitive needs;
• cultivate independence in work and accuracy;
• develop mutual understanding and self-confidence.

Type of lesson: Repetition and generalization of the studied material.

Equipment and use of educational material.

1. “Informatics in games and tasks.” 3rd grade in 2 parts. Textbook-notebook, part 2. Team of authors Goryachev A.V., Gorina K.I., Suvorova N.I. – M.: “Balass”, 2008.

2. Handouts. Worksheet assignments. Appendix 2.

3. Personal computer. Application package "Graphical editor Paint".

4. Multimedia projector.

5. Interactive whiteboard and SmartBoard software. Presentation "Sets. Relations between sets." Annex 1.

6. A set of numbers from 1 to 5 for each student (it is desirable that each number has its own color).

During the classes

I. Organizational moment

II. Repetition and generalization of material.

Working with an interactive whiteboard

1 page. Topic title.

Page 2. Multitudes. Elements of a set.

Oral work (teacher asks questions and students answer)

What is a set? ( group of objects with a common name).

What are sets made of? (from elements).

Give an example of an empty set (many tails for people, many arms for animals, ......); sets with one element (many letters K in the Russian alphabet, human heads, ......).

What sets are shown in the picture? How many elements does this set contain? (many houses - three elements, many buckets - one element, many trees - many elements, many flowers - many elements, many stones - eight elements,......).

So tell me, how many elements can a set include? ( a set may include one element, may include many or not very many elements, and may be empty - this is a set in which there is no element).

The activities on pages 3-6 are completed simultaneously on the board and on the worksheets. Students take turns going to the board.

Page 3. Multitudes. Subsets.

Orally.

What is the name of a set that is part of another set? (subset).

Working with an interactive whiteboard.(three students come to the board in turn and shade the circles with a stylus).

To complete this task, students must find the symbol for each set in the table, determine which set contains more elements, and fill in the large circles.

• First student: There are more children than third graders and schoolchildren, so we paint the largest circle in red.
• Second student: There are more schoolchildren than third graders, so we paint the middle circle blue.
• Third student: There are fewer third graders than schoolchildren and children, so we paint the smallest circle green.

application) and fill in the circles using colored pencils.

Page 4. Intersection of many.

Orally.

What sets are called intersecting? (if they have common elements).

Exercise: Distribute elements into appropriate sets.

Students take turns going to the board and moving elements into the corresponding sets, and they are required to explain why he distributes a given element to a specific set.

For example: watermelon - edible, but not red - many edible; pepper - edible and red - intersection of sets; dress - red, but not edible - lots of red; the ball - not edible and not red - is located outside the sets.

The rest of the students work on the worksheets (see application) and show the path of movement using an arrow.

5 page. Mutual arrangement of sets.

Second student: Lots of wild animals and lots of domestic animals. These sets have the same elements (for example, a pig, a duck, a goose - a domestic animal and a wild one), which means they intersect. Let's connect with the first circuit.

Third student: Lots of birds and lots of insects. There are no birds that are insects and there are no insects that are birds, which means the sets do not intersect. We connect with the third circuit.

Exercise: Establish a correspondence between the scheme and sets.

6 page. Multitudes. Elements of a set. Intersection and union of sets (Words “NOT”, “AND”, “OR”).

Exercise: Enter the numbers of the figures in the figures. How many squirrels are there in each set? (Write your answers in the cells of the table). Color in the parts of the figures in the table.

Student answers:

Squirrel in Figure 9.

Squirrel with mushrooms 3.

Squirrel with nuts 4.

Squirrel with mushrooms and nuts 1 (Fig. 9). In the table, the area of ​​intersection of the circle and the oval is shaded; on the diagram, the number 9 is written in the area of ​​intersection.

Squirrel with mushrooms or nuts 6 - these are squirrels that have both mushrooms and nuts (Fig. 9), only nuts (Fig. 3,7), only mushrooms (Fig. 1, 4, 6). In the table, the entire circle and the entire oval are shaded. On the diagram, the numbers 3, 7 are written in a circle, outside the oval; in the oval outside the circle - numbers 1,4, 6.

Squirrels that do not have mushrooms 6 (Fig. 1, 2, 4, 5, 6, 8). In the table, only the circle area is not shaded.

Squirrels that do not have nuts 5 (Fig. 2, 3, 5, 7, 8). In the table, only the oval area is not painted over.

On the diagram, the numbers 2, 5, 8 are written in a rectangle, outside the circle and oval - these are squirrels that do not have nuts and mushrooms.

III. Physical education minute

The robot does exercises and counts in order:

Once! - the contacts do not spark,
- Two! - joints don’t creak,
- Three! - the lens is transparent.
I am correct and beautiful!

1,2,3,4,5 - We can get down to business!

IV. Knowledge control. Independent work.

Students in the class are divided into two groups.

Group 1 completes tasks on pieces of paper Appendix 3, Group 2 performs tasks on computers Appendix 4. After 5-7 minutes, students change places.

The task on the pieces of paper is done using colored pencils.

1 task. Using geometric shapes, a rectangle and a circle, depict the proposed situation.

Task 2. Color in part of the diagram so that the statement is true.

The task on computers is performed in the Paint graphic editor. The first and second tasks are presented in one file.

The path to the file ( the teacher speaks and the students follow his commands).

Desktop -> Folder 3rd class -> (double-click to open) -> File Homework -> (right-click) -> Open with Paint.

1 task. Using geometric primitives, a rectangle and an ellipse, depict the proposed situation.

Task 2. Using the Fill tool, paint over part of the diagram so that the statement is true.

After completing the tasks, the teacher checks the correctness of the work.

V. Lesson summary.

Guys, today we repeated what a set, subset, intersection and union of sets are.

• So tell me, how many elements can there be in a set? (as many as you like).
• What is the name of a set that is part of another set? (subset).
• And what elements are included in the intersection of two sets? (which are included in both one and the other set).

VI. Homework.

1 task presented on pieces of paper and distributed to each student (see. application). Color in the parts of the figures in the table. Look in the table to see how many hedgehogs there should be in each set. Color the hedgehogs. Write the numbers in the empty cells of the table.

2 task performed at the request of the student. Come up with a task on the relative position of sets. Prepare your work on A4 sheets. The work must contain the name of the sets, diagrams, and drawings.

VII. Reflection.

• What task did you enjoy most today?
• What task caused difficulty?

Each of you has a set of natural numbers from 1 to 5 on your desk; hang one of the numbers, at which mark you rate the lesson, on the mood tree.

Mathematical analysis is the branch of mathematics that deals with the study of functions based on the idea of ​​an infinitesimal function.

The basic concepts of mathematical analysis are quantity, set, function, infinitesimal function, limit, derivative, integral.

Size Anything that can be measured and expressed by number is called.

Many is a collection of some elements united by some common feature. Elements of a set can be numbers, figures, objects, concepts, etc.

Sets are denoted by uppercase letters, and elements of the set are denoted by lowercase letters. Elements of sets are enclosed in curly braces.

If element x belongs to the set X, then write x∈X (∈ - belongs).
If set A is part of set B, then write A ⊂ B (⊂ - contained).

A set can be defined in one of two ways: by enumeration and by using a defining property.

• A=(1,2,3,5,7) - set of numbers
• Х=(x 1 ,x 2 ,...,x n ) - set of some elements x 1 ,x 2 ,...,x n
• N=(1,2,...,n) — set of natural numbers
• Z=(0,±1,±2,...,±n) — set of integers

The set (-∞;+∞) is called number line, and any number is a point on this line. Let a be an arbitrary point on the number line and δ be a positive number. The interval (a-δ; a+δ) is called δ-neighborhood of point a.

A set X is bounded from above (from below) if there is a number c such that for any x ∈ X the inequality x≤с (x≥c) holds. The number c in this case is called top (bottom) edge set X. A set bounded both above and below is called limited. The smallest (largest) of the upper (lower) faces of a set is called exact top (bottom) edge of this multitude.

Basic number sets

If a set does not contain a single element, then it is called empty set and is recorded Ø .

Elements of logical symbolism

Notation ∀x: |x|<2 → x 2 < 4 означает: для каждого x такого, что |x|<2, выполняется неравенство x 2 < 4.

Quantifiers are often used when writing mathematical expressions.

Quantifier is called a logical symbol that characterizes the elements following it in quantitative terms.

• ∀- general quantifier, is used instead of the words “for everyone”, “for anyone”.
• ∃- existence quantifier, is used instead of the words “exists”, “is available”. The symbol combination ∃! is also used, which is read as if there is only one.

Set Operations

Two sets A and B are equal(A=B) if they consist of the same elements.
For example, if A=(1,2,3,4), B=(3,1,4,2) then A=B.

By union (sum) sets A and B is a set A ∪ B whose elements belong to at least one of these sets.
For example, if A=(1,2,4), B=(3,4,5,6), then A ∪ B = (1,2,3,4,5,6)

By intersection (product) sets A and B is called a set A ∩ B, the elements of which belong to both the set A and the set B.
For example, if A=(1,2,4), B=(3,4,5,2), then A ∩ B = (2,4)

By difference The sets A and B are called the set AB, the elements of which belong to the set A, but do not belong to the set B.
For example, if A=(1,2,3,4), B=(3,4,5), then AB = (1,2)

Symmetrical difference sets A and B is called the set A Δ B, which is the union of the differences of the sets AB and BA, that is, A Δ B = (AB) ∪ (BA).
For example, if A=(1,2,3,4), B=(3,4,5,6), then A Δ B = (1,2) ∪ (5,6) = (1,2,5 ,6)

Properties of set operations

A ∪ B = B ∪ A
A ∩ B = B ∩ A

(A ∪ B) ∪ C = A ∪ (B ∪ C)
(A ∩ B) ∩ C = A ∩ (B ∩ C)

Countable and uncountable sets

In order to compare any two sets A and B, a correspondence is established between their elements.

If this correspondence is one-to-one, then the sets are called equivalent or equally powerful, A B or B A.

The set of points on the leg BC and the hypotenuse AC of triangle ABC are of equal power.

A set can be defined by listing its composition separated by commas in curly brackets. For example, if the set consists of the numbers 5, 7 and 25, then write . The numbers themselves 5, 7, 25 are called elements of the set. The order in which the elements of the set are listed in brackets does not matter. A set cannot contain the same element twice. The fact that 5 is an element of the set is written as follows: . A set that does not have a single element is called empty and is denoted by .

Two sets are said to be equal if they consist of the same elements. For example, if , then .

If all the elements of a set are contained in the set, then they say that the set is a subset of the set and write. For example, the set is a subset of the set described above. The empty set is a subset of any set. Moreover, each set is a subset of itself: .

A number of operations can be performed on sets.

Union of sets

Drawing. Union of sets
A set is a union of sets and if it includes all the elements of the set and all the elements of the set . The union of sets is written as follows: . Let us explain this by depicting sets and using Euler circles (Fig. 1). Each of the sets is depicted using circles. The set in Fig. 1 is shown as a shaded figure. Let , . Then .

For any set the statement is true

Intersection of many

A set is the intersection of the sets and if it contains only those elements that belong to both the set and the set. Notation for the intersection of sets: . For the sets mentioned above.

Drawing. Intersection of many
Here's another example. . Here the intersection of sets is an empty set, because The sets have no common elements.

Drawing. Set difference
Set difference

The difference of sets is the set of those elements from that are not contained in . The difference between sets is denoted as follows:

For the sets already mentioned. In Figure 3, the set difference is shaded.

Symmetric set difference

Denoted by . As shown in Figure 4 in red,

The statement is also true

Drawing. Symmetric set difference

In other words, the symmetric difference of sets consists of all those elements of the first set that are not in the second, together with those elements of the second set that are not in the first. For the sets from the previous examples .

Sets in Delphi and FreePascal

FreePascal and Delphi support data types for working with sets. The set description format is as follows

Type type_name = set of base_type

Sets in Pascal consist of data of the same ordinal type, called base. A base type can have a maximum of 256 different values. The number of elements of the set cannot be more than 255.

Examples of set descriptions

Type Dgt = 0..9;

Digits = set of Dgt;

DigitChar = set of "0".."9";

The top line of the example contains the definition of the range type Dgt, the second line defines the type Digits, which is a set of elements of the base type Dgt. It was possible to do without a separate declaration of a range type. For example, the DigitChar type represents a set of characters, each of which can be in the range from "0" to "9".

The base type does not have to be a range type. The following defines a set of elements of type Char. This is acceptable because the Char type contains 256 different values.

Type Junk = Set of Char;

However, using Integer as the base type would be a mistake because the number of possible values ​​of this type is greater than 256:

Type Junk = Set of Integer ; //It is forbidden!!!

It is unacceptable to use it as a base type when describing sets and real data types, for example real, since they are not ordinal.

After defining the type of a set, you can describe variables of this type. For example,

You can use the design set of and right when declaring variables. For example,

Var sc: set of 0..9;

Creating Sets

To create a set, use the so-called set constructor. It can be written in the following ways.

1. 
   The elements of the set are listed in square brackets, separated by commas. They must be constants, variables, or expressions of a base type. For example, sc:=, where x- a variable of integer type.
2. 
   [a..b]. In this case, the set contains all values ​​of the base type, starting with a and ending b. With this method of specifying the set there should be a b. For example, the expression sc:= means the same as sc:=.
3. 
   A combination of methods 1 and 2. For example, sc:=.
4. 
   The empty set is specified by an open and immediately closed square bracket. For example, sc:=.

Examples of problem solving

Problem 1

Solution

Let M- the set of all lowercase English letters from a before z. Let us denote by B a set of lowercase English letters already found when viewing from the beginning of the line.

We can propose such an algorithm.

Let's write a program.

Program EngLetter;

i, len: Integer;

B, M: set of Char;

If s[i] in B Then
WriteLn("Yes");
End;

If s[i] in M ​​Then

B:=B+]; //Combining sets

WriteLn("No");

Problem 2

Solution

Let be the set of digits of a number, and let be the set of digits of a number. Then the set of digits that are both in the notation of the number and in the notation of the number ,

If , then there are general numbers. Each of the described sets contains no more than 10 elements, each element no more than 10. This means that Pascal language sets can be used to represent them.

Let's define data types

Type Digit = 0..9;

SetDigit = set of Digit;

Let us highlight the subtask of constructing a set of digits of a natural number x into the procedure

Then we can propose the following algorithm for solving the problem.

We will create a program using this algorithm.

Type Digit = 0..9;

SetDigit = set of Digit;

Procedure MakeSet(x: Integer; out s: SetDigit);

Var last: Digit;

s:=; //We haven't found a single digit of x yet

While x>0 Do
last:= x mod 10; //Last digit of number x

s:=s+; //Include last in the set of digits of the number x

x:=x div 10 //Unhook the last digit

Var m,n,s,r: Integer;

WriteLn("sum",s);

WriteLn("difference",r);

WriteLn("No common digits")

WriteLn("There are general numbers")

Questions and tasks for independent solution

1. 
   Calculate without a computer
   1. 
      d:=+;
   2. 
      c:=*;
   3. 
      c:= – ;
   4. 
      x:=10 in ;
2. 
   Is it possible to use ShortInt as a base type when describing a set? Byte? Int64? Char? String? Double?
3. 
   Write a program to solve the problem. How many odd digits are there in the string entry? s? Count each digit as many times as it appears in the line. For example, in the line "AwDc12 h215" there are three odd digits: two ones and a five.
4. 
   The line contains text in Russian, written in capital letters. Print those vowels that are not in this text.
5. 
   Determine which characters in a string b not in line a. For example, if a="abcd", b="baMCc", the answer will be "MC".
6. 
   Determine the common digits in the notation of natural numbers a And b, i.e. numbers that are also in the number record a, and in the notation of numbers b. Is it true that the number c recorded only using these common for a And b numbers provided that the numbers can be reused?
7. 
   At the end of the sentence, one of the punctuation marks is placed: a period, a question mark, an exclamation mark - or a combination of them, for example, three dots in a row, a question mark with an exclamation mark, several exclamation marks in a row. Write a program to count the number of sentences in a given string. There are no spaces between consecutive punctuation marks.

Literature

1. 
   Michael van Canneyt. Reference guide for Free Pascal, version 2.4.2. -November 2010
2. 
   Borland Help for BDS2006.
3. 
   Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis.: Textbook for universities. - M.: Nauka, 1989.
4. 
   Cormen T., Leiserson Ch., Rivest R., Stein K. Algorithms. Construction and analysis. Second edition. - Moscow, St. Petersburg, Kyiv. Williams Publishing, 2010.
5. 
   A bunch of. // http://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE
6. 
   Faronov V.V. Turbo Pascal 7.0. Beginner course. Tutorial. - M.: “Knowledge”, 1998