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Write down the sets that are shaded. Sets

Write down the sets that are shaded. Sets

26.07.2023

Goals and objectives of the lesson:

Educational:

repeat and consolidate the received representations:
about a set, an element of a set, a subset, an intersection of sets, a union of sets;
consolidate skills:
determine the belonging of elements to a set and its subset (subsets), as well as to a set that is an intersection, a 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, which is the intersection, union of sets and name the elements from this area;
determine the nature of the relationship between two given sets (set-subset, have intersection, have no intersection);
correctly depict the proposed situation;
computer skills in the graphic editor Paint.

Developing:

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

Educators:

foster friendly relations in the student team;
educate the cognitive need;
to cultivate independence in work, accuracy;
develop mutual understanding and self-confidence.

Lesson type: Repetition and generalization of the studied material.

Equipment and use of educational material.

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

2. Handout. Worksheet assignments. Appendix 2

3. Personal computer. Application package "Graphic 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 the material.

Working with the interactive whiteboard

1 page. Topic title.

2 page. Sets. Set elements.

Oral work (teacher asks questions and students answer)

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

What are sets made of? (from elements).

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

What sets are shown in the figure? How many elements are in this set? (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 not a single element).

The tasks on page 3-6 are completed simultaneously on the board and on the worksheets. Students take turns going to the blackboard.

3 page. Sets. Subsets.

Orally.

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

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

To complete this task, students must find the designation of 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 over the largest circle in red.
Second student: There are more schoolchildren than third graders, so we paint the middle circle in blue.
Third student: There are fewer third-graders than schoolchildren and children, so we paint over the smallest circle in green.

application) and fill in the circles with colored pencils.

4 page. Intersection of many.

Orally.

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

Exercise: Distribute the elements to the appropriate sets.

Students take turns going to the board and moving the elements to the appropriate sets, while it is required to explain why he distributes this element to a specific set.

For example: watermelon - edible, but not red - a lot of edible; pepper - edible and red - the intersection of sets; dress - red, but not edible - a lot 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 to move with an arrow.

5 page. Mutual arrangement of sets.

Second student: Lots of wild animals and lots of pets. 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. We connect with the first scheme.

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

Exercise: Establish correspondence between schema and sets.

6 page. Sets. Set elements. Intersection and union of sets (Words "NOT", "AND", "OR").

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

Student responses:

Squirrels in figure 9.

Squirrels with mushrooms 3.

Squirrels with nuts 4.

Squirrels with mushrooms and nuts 1 (Fig. 9). In the table, the area of the intersection of the circle and the oval is painted over; in the diagram, the number 9 is written in the intersection area.

Squirrels with mushrooms or nuts 6 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 painted over. On the diagram in a circle, outside the oval, the numbers 3, 7 are written; in the oval outside the circle - the numbers 1,4, 6.

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

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

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

III. Physical education minute

The robot does exercises and counts in order:

One! - contacts do not spark,
- Two! - the joints do not creak,
-Three! - the lens is transparent.
I am fit and beautiful!

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

IV. Knowledge control. Independent work.

The students in the class are divided into two groups.

1 group performs tasks on the sheets Appendix 3, Group 2 performs tasks on computers Appendix 4 After 5-7 minutes, students change places.

The assignment is done on paper using colored pencils.

1 task. With the help of geometric shapes, a rectangle and a circle depict the proposed situation.

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

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

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

Desktop -> Grade 3 Folder -> (double-click open)-> Self-Work File -> (right-click)-> Open with Paint.

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

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

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

V. The results of the lesson.

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

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

VI. Homework.

1 task presented on leaflets and distributed to each student (see. application). Color in the table parts of the figures. Look in the table how many hedgehogs 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 for the mutual arrangement of sets. Submit your work on A4 paper. The work should contain the name of the sets, diagram, drawings.

VII. Reflection.

What task did you enjoy the most today?
What task caused the problem?

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

The concept of a set is one of the basic concepts of mathematics. There is no definition for it. The English mathematician Bertrand Russell described this concept as follows: "A set is a collection of various elements, conceived as a single whole." We can talk about the set of faces of a polygon, the set of points of a straight line, the set of natural numbers, the set of letters of the Russian alphabet, etc.

A set can be specified by listing its composition separated by commas in curly braces. For example, if the set consists of numbers 5, 7 and 25, then write . The numbers 5, 7, 25 themselves 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 any elements is called empty and 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 the set is said to be 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. In addition, each set is a subset of itself: .

You can perform a number of operations on sets.

Union of sets

Drawing. Union of sets
A set is a union of sets and if it includes all elements of the set and all elements of the set . The union of sets is written like this: Let us explain this by depicting sets and using Euler circles (Fig. 1). Each of the sets and 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. Set intersection notation: . For the sets mentioned above.

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

Drawing. Set difference
Set difference

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

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

Symmetric set difference

Designated . 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 sets from previous examples .

Sets in Delphi and FreePascal

Defining types and declaring variables

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

Type typename = set of base_type

Sets in Pascal consist of data of the same ordinal type, called base. The base type can have no more than 256 distinct values. The number of elements in the set cannot exceed 255.

Examples of set declarations

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 the set of elements of the base type Dgt. It was possible to do without a separate declaration of the range-type. For example, the DigitChar type is a set of characters, each of which can range from "0" to "9".

The base type need not be a range type. The set of elements of type Char is defined below. This is legal because the Char type contains 256 distinct values.

Type Junk = Set of Char;

However, using Integer as the base type would be an error because the number of possible values for this type is greater than 256:

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

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

Once the type of the set is defined, variables of this type can be declared. For example,

You can use the design set of and right at the declaration of variables. For example,

Varsc: set of 0..9;

Creating sets

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

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

Operations on sets

Operator	Description	Example
+	Union of sets	c:=a+b; d:=+;
*	Intersection of many	c:=*;
-	Set difference	c:= - ;
=	Checking the equality of sets. The result is of type Boolean	Program Sample1; x:==;
	Atrue, if it is.	Program Sample2; Var a,b: set of 1..100; a:=;
in	Boolean expression x in A checks if x set element A. Variable (or constant) x should be the base for the set A type.	x:=10 in ;
>	Symmetric difference of sets. Only for free pascal . IN Delphi does not work. In the example, all elements of the set C, which is the symmetric difference of the sets A and B, are displayed on the screen. There is another way to find out the composition of the set, except for using the operator in, No.	($mode delphi) Program Sample4; Var a,b,c: set of Byte; b:=; For i:=0 To 255 Do
	Checking the inequality of sets. AB matters true if set A is not equal to set B.	($mode delphi) Program Sample5; Var a,b: set of Byte; b:=;

Examples of problem solving

Task 1

Is there a line s at least two identical lowercase English letters? (For example, the string "book" has such letters. This is the letter "o". But the string "Elem 1221" does not.)

Solution

Let M- the set of all lowercase English letters from a before z. 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.

If we have reached point 5 of the algorithm, then there is not a single lowercase English letter in the string.

Let's write a program.

Program EngLetter;

i, len: Integer;

B, M: set of Char;

WriteLn("Enter string");
len:=length(s);
While iBegin

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

If s[i] in M Then

B:=B+]; // Union of sets

end;

WriteLn("No");

Task 2

Given natural numbers and . (

) Does the decimal notation of natural numbers have the same digits?

Solution

Let be the set of digits of , and be the set of digits of . 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 common 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.

Define data types

Type Digit = 0..9;

SetDigit = set of Digit;

We single out the subproblem of constructing the set of digits of a natural number x into the procedure

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

Now we will compose the algorithm of the MakeSet procedure.

What does the expression "at least one digit remains in the record of a number" mean? Finding partial quotients from dividing by 10, we will eventually get zero.

Let's 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:=; // Haven't found any digits 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 x

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

end;

Varm,n,s,r: Integer;

Write("m, n = ");
MakeSet(s,A);

WriteLn("sum ",s);

WriteLn("difference", r);

WriteLn("There are no common digits")

WriteLn("There are total numbers")

Questions and tasks for independent solution

Calculate without a computer
1. d:=+;
2. c:=*;
3. c:= - ;
4. x:=10 in ;
Is it possible to use ShortInt as the base type when describing a set? Bytes? Int64? Char? String? Double?
Write a program to solve the problem. How many odd digits are there in a string s? Count each digit as many times as it occurs in the string. For example, in the string "AwDc12 h215" there are three odd digits: two ones and five.
The line contains text in Russian written in capital letters. Output those vowels that are not in this text.
Determine which characters of a string b not in line a. For example, if a="abcd", b="baMCc", the answer is "MC".
Determine the common digits in the notation of natural numbers a And b, i.e. digits that are also in the number entry a, and in the notation of the number b. Is it true that the number c written only using these common a And b digits, provided that the digits can be reused?
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 point, 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

Michael van Canneyt. Reference guide for Free Pascal, version 2.4.2. - November 2010
Borland Help for BDS2006.
Kolmogorov A.N., Fomin S.V. Elements of the theory of functions and functional analysis.: A textbook for universities. - M.: Nauka, 1989.
Kormen T., Leyzerson Ch., Rivest R., Stein K. Algorithms. Construction and analysis. Second edition. - Moscow, St. Petersburg, Kyiv. Publishing house "Williams", 2010.
A bunch of. // http://en.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B6%D0%B5%D1%81%D1%82%D0%B2%D0%BE
Faronov V.V. Turbo Pascal 7.0. Initial course. Tutorial. - M.: "Knowledge", 1998

Mathematical analysis is a 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.

Value everything that can be measured and expressed by a number is called.

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

Sets are denoted by capital letters, and elements of a set by lowercase letters. Set elements 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 (⊂ - is contained).

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

For example, the enumeration defines the following sets:

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

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

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

Basic Numeric Sets

N	(1,2,3,...,n) The set of all
Z	(0, ±1, ±2, ±3,...) Set whole numbers. The set of integers includes the set of natural numbers.
Q	A bunch of rational numbers. In addition to whole numbers, there are also fractions. A fraction is an expression of the form , where p is an integer, q- natural. Decimals can also be written as . For example: 0.25 = 25/100 = 1/4. Integers can also be written as . For example, in the form of a fraction with a denominator of "one": 2 = 2/1. Thus, any rational number can be written as a decimal fraction - finitely or infinitely periodic.
R	Many of all real numbers. Irrational numbers are infinite non-periodic fractions. These include: Together, two sets (rational and irrational numbers) form the set of real (or real) numbers.

If a set contains no elements, then it is called empty set and recorded Ø .

Elements of logical symbolism

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

quantifier

When writing mathematical expressions, quantifiers are often used.

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

∀- general quantifier, is used instead of the words "for all", "for anyone".
∃- existential quantifier, is used instead of the words "exists", "has". The symbol combination ∃! is also used, which is read as there is only one.

Operations on sets

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.

Union (sum) sets A and B is called the 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)

Intersection (product) sets A and B is called a set A ∩ B, whose elements 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)

difference sets A and B is called a 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)

Symmetric 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)