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The concept of set. Georg Cantor (1845-1918) Professor of mathematics and philosophy, founder of modern set theory. “By plurality we mean the unification into a whole of certain objects of our representation or thought that differ from each other.” Georg Cantor
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The concept of set. The basic concept in mathematics is the concept of set. The concept of set refers to initial concepts that cannot be defined. By set we mean a certain collection of homogeneous objects. The items (objects) that make up a set are called elements.
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Set designation Sets are denoted by capital letters of the Latin alphabet: A, B, C, X, etc. Elements of a set are denoted by lowercase letters of the Latin alphabet: a, b, c, d, etc. The notation M = (a, b, c, d) means that the set M consists of elements a, b, c, d. Є – a sign of belonging. The notation a є M means that the object a is an element of the set M and reads like this: “a belongs to the set M”
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Number of a set Number of a set is the number of elements in a given set. It is denoted as follows: n Written as follows: n (M) = 4 There are sets: Finite sets - consist of a finite number of elements, when all elements of the set can be counted. Infinite sets - when it is impossible to count all the elements of the set. Empty sets are sets that do not contain elements and are denoted as follows: Ø. Write it like this: n (A)=0 ; A= Ø The empty set is a subset of any set.
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Types of sets: Discrete sets (discontinuous) - have separate elements. This way the bills are recognized. Continuous sets - no separate elements. Recognized by measurement. Finite sets consist of a finite number of elements when all elements of the set can be counted. Infinite sets - when it is impossible to count all the elements of the set. Ordering sets. An element of a set precedes or follows another. The set of natural numbers arranged in a natural series. Unordered sets. Any unordered set can be ordered.
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Methods for defining sets By enumerating elements (suitable for finite sets). Indicate the characteristic property of the set, i.e. a property that all elements of a given set have. Using an image: On a ray In the form of a graph Using Euler circles. Mainly used when performing operations on sets or demonstrating their relationships.
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Subset If any element of set B belongs to set A, then set B is called a subset of set A. - Inclusion sign. Notation B A means that set B is a subset of set A.
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Types of subsets Own subset. A set B is called a proper subset of a set A if the following conditions are met: В≠Ø, В≠А. Not proper subsets. A set B is called a non-proper subset of a set A if the following conditions are met: B≠Ø, B=A. The empty set is a subset of any set. Any set is a subset of itself.
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A B A=B Set Equalities Sets are equal if they consist of the same elements. Two sets are equal if each is a subset of the other. In this case they write: A=B
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Operations on sets Intersection of sets. Union of sets. Difference of sets. Complement of a set.
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Union of sets The union of sets A and B is the set of all objects that are elements of set A or set B. U is a union sign. A U B reads like this: “The union of set A and set B.”
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Intersection of sets The intersection of sets A and B is a set containing only those elements that simultaneously belong to both set A and set B. The ∩-sign of intersection corresponds to the conjunction “and”. A ∩ B reads like this: “Intersection of sets A and B”
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Difference of sets The difference of sets A and B is the set of all objects that are elements of set A and do not belong to set B. \ is the difference sign, corresponds to the preposition “without”. The difference between sets A and B is written as follows: A \ B
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Complement of a set The set of elements of set B that do not belong to set A is called the complement of set A to set B. Often sets are subsets of some basic or universal set U. The complement is denoted by Ā
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Properties of sets The intersection and union of sets have the following properties: Commutativity Associativity Distributivity
“Elements of a set” - Sets are usually denoted by capital letters of the Latin alphabet: A, B, C... Elements of a set are usually denoted by lowercase letters of the Latin alphabet: a, b, c... Relationships between sets are visually represented using Euler circles. The empty set is considered a subset of any set. If a set does not contain any elements, it is called empty and is denoted by? or 0.
“Elements of the set” - Characteristic features. List. Lots of sparrows. Examples. Description. Subset. The description includes the main, characteristic feature of the set. Actions with sets. Complement of a set. Universal set. Multitudes. Georg Cantor. Infinite sets cannot be specified as a list. Methods for specifying sets.
“Intersection and union of sets” - Some sets X and Y do not have common elements. The sets A and B are depicted in circles in the figure. 1. Intersection of sets. For example: X is the set of prime numbers not exceeding 25; Y is a set of two-digit numbers not exceeding 19. The figure formed by the intersection of circles, shaded in the figure, depicts the set C.
“Sets and operations on them” - The cardinality of a set is a set with a finite number of elements. The Cartesian (direct) product of the sets A and B is the set of ordered pairs. Multitudes. The complement of set C is the complement of set B, which consists of elements of set A that are not included in set B. Sets are written in various forms: 1) in curly brackets by simple enumeration: A = (1,2,3) 2) graphically.
“Comparison of sets” - Practical work on the computer. Work in a notebook. Comparison of sets. Physical education minute. Lots of Insects. Graphic dictation. We teach computer science We will gain a lot of knowledge Think, think head We study sets Hands up and one, two, three And now bend down Come on, fish, show yourself Turns to the right, to the left Sit down and get down to business.
“Set Theory” - Thus, we have performed the operations of intersection, union and difference of two sets. It is designated A’ or A and reads “not A”. Basic numerical sets. It is also believed that the empty set is a subset of any set. The concept of set. Definition. How many students can skate and ski?
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Slide captions:
Equal sets. Empty set. Ø sign. 3rd grade. Mathematics Peterson L.G. http://aida.ucoz.ru
Compare the elements of the sets in the first and second rows. Is there an element in the first row that is not in the second? Is there an element in the second row that is not in the first? http://aida.ucoz.ru
Compare the sets in the top and bottom rows. Which row has an extra element?
Two sets are equal if they contain the same elements. If sets A and B are equal, then write A = B, and if they are not equal, then write A ≠ B. Example: Let A = (raspberry; strawberry; currant), B = (strawberry; raspberry; currant), C = (currant; raspberry; cherry), D = (raspberry; strawberry; currant; gooseberry). A = B (they have the same elements, just in a different order); A ≠ C (in A there is a strawberry, and in C there is a cherry instead); A ≠ D (in D the extra element is gooseberry).
Is the equality written correctly? Why? ( ; ; ; ; ; ) = ( ; ; ; ; ; ) ; ; YES, NO ( ; ; ; ) = ( ; ; ) ; YES, NO ( ; ; ; ) = ( ; ; ; ) ; ; YES, NO
Let A = (0; 1; 2). Which of the sets B = (2; 0; 1), C = (1; 0), D = (3; 2; 1; 0) are equal to set A, and which are not equal to it? Explain how to write it down. A A A B C D = ≠ ≠
How many elements does it contain: Many days of the week? Lots of desks in the front row? Many letters of the Russian alphabet? Does Murka the cat have many tails? Does Petya have many noses? Lots of horses grazing on the moon? If a set has no elements, then it is said to be empty. The empty set is denoted as follows: Ø. Come up with some examples of an empty set.
Homework assignment. We are working in the textbook. No. 11,12 page 9
On the topic: methodological developments, presentations and notes
This lesson was developed based on the textbook “Computer Science in Games and Problems” by A.V. Goryacheva. This lesson, the fourth in a series of lessons on the topic "Multiple", is a lesson in summarizing and consolidating the knowledge gained on...
A bunch of. Subset. Intersection of many. (We resettle multitudes)
· To consolidate ideas about sets, subsets, the intersection of two sets. · To consolidate the ability to define...
Equal sets.
Pedagogical
target
Introduce the concept of “equal sets”; learn to distinguish between sets, combine objects into groups based on similar characteristics, and isolate individual objects from a group.
Type, type of lesson
Lesson in learning new knowledge
Planned
results
(subject)
Form and compare sets; name the elements of a set; distinguish between equal and unequal sets. Correctly use mathematical concepts in speech.
Universal
educational
actions
Personal: awareness of the mathematical components of the surrounding world.
Metasubject:
Regulatory: mastering ways of combining objects and separating them from a group according to certain characteristics.
Cognitive: understanding the concept of “equal sets” at a subject-specific level.
Communicative: ability to use simple speech means; engage in dialogue with the teacher and peers, in collective discussion; answer teacher's questions.
Forms and methods
training
Shapes: frontal, individual, pair work
Methods: verbal, visual, practical
Basics
content of the topic, concepts and terms
A bunch of. Elements of a set. Equal sets.
Set, element of set
Educational Resources
Dorofeev G.V., Mirakova T.V. Mathematics: Textbook: 1st grade, 1st part; – M.: Education, 2014.
Dorofeev G.V., Mirakova T.V. Mathematics: Workbook: 1st grade, part 1.. - M.: Prosveshchenie, 2014.
Dorofeev G.V., Mirakova T.V. "Mathematics. Methodological recommendations. 1st grade. Federal State Educational Standard" - M.: Education, 2011.
Electronic supplement to the textbook by G. V. Dorofeev, T. N. Mirakova (CDpc)" - M.: Prosveshchenie, 2014.
During the classes.
I. Organizational moment
II. Updating knowledge
Today, together with Anya and Vanya, we will go for a walk in a forest clearing. Look how beautiful it is!
How to call in one word the objects that are shown in the picture?(flowers).
What is a group of objects called in mathematics?(A bunch of)
- What is a single object of a set called?(element)
Name the elements of many colors.(chamomile, cornflower, bell, tulip, rose)
- How many groups can we divide this set into? Which?(1: chamomile, 2: bell and cornflower, 3: rose and tulip)
By what property did we divide the set?(By color)
Let's count the number of elements of the set from right to left, from left to right.(counting items)
How many elements of the set of colors are there? (5)
Let's test your memory. What number is the bell?(third)
What flower is to the right of it? (tulip) In what place?(on the fourth)
What flower is to the left of the bell?(cornflower) Where?(on the second)
How much is a rose worth?(fifth, last)
What flower is to the right of the daisy?(cornflower)
Which flower is between the cornflower and the rose?(bell, tulip)
III. Formulation of the problem. Discovery of new knowledge.
While we were looking at the flowers and training our memory, Anya and Vanya picked bouquets for their mothers. Did they get the same bouquets? (No). Can we name many bouquets?equal ? (?)
Today in the lesson we will learn which sets are called equal.
Let's listen to our expert, Professor Samovarov.
After the first part of the video we conclude:If sets consist of the same elements, then they are equal.
After the second part of the video we conclude:If sets differ in at least one element, then they are not equal.
Let's return to Anya and Vanya. Let's answer it. Can we name the many bouquets of Anya and Vanya?equal ? (No).
Physical education minute.
IV. Consolidation of knowledge
Working in a workbook. Page 28 No. 1
Let's compare the sets in orange frames. Are they equal? (yes, the elements in them are the same )
= )
Let's compare the sets in blue frames. Are they equal? (no, because in the right set there is a pumpkin, and in the left set there is a watermelon)
What sign should we put between these sets? ("not equal" sign/cross out the "equal" sign )
Let's compare the sets in green frames. Are they equal? ? (yes, the elements in them are the same )
Let's compare the sets in pink frames. Are they equal? (no, because in the right set there is a small blue square and a large yellow circle, and in the left set there is a large yellow square and a small blue circle)
Work in pairs.
Now you will work in pairs. Boys should draw a lot of squares on their half of the sheet, and girls should draw a lot of triangles on their half of the sheet. Agree on the number of elements. Your sets must be equal.
Work according to the textbook.Page 34 No. 1
V. Lesson summary. Reflection.
What new knowledge did we gain in class today?
– What did you like most about the lesson?
Raise a blue pencil if the topic of the lesson is clear to you and you can easily determine whether sets are equal, a red pencil if you have difficulties and need to work on this topic.
Slide 2
Compare the elements of the sets in the first and second rows. Is there an element in the first row that is not in the second? Is there an element in the second row that is not in the first?
http://aida.ucoz.ru
Slide 3
Compare the sets in the top and bottom rows. Which row has an extra element?
Slide 4
Two sets are equal if they contain the same elements. If sets A and B are equal, then write A = B, and if they are not equal, then write A ≠ B.
Example: Let A = (raspberry; strawberry; currant), B = (strawberry; raspberry; currant), C = (currant; raspberry; cherry), D = (raspberry; strawberry; currant; gooseberry). A = B (they have the same elements, just in a different order); A ≠ C (in A there is a strawberry, and in C there is a cherry instead); A ≠ D (in D the extra element is gooseberry).
Slide 5
Is the equality written correctly? Why?
( ; ; ; ; ; ) = ( ; ; ; ; ; ) ; ; YES, NO ( ; ; ; ) = ( ; ; ) ; YES, NO ( ; ; ; ) = ( ; ; ; ) ; ; YES, NO
Slide 6
Let A = (0; 1; 2). Which of the sets B = ( 2; 0; 1), C = ( 1; 0), D = ( 3; 2; 1; 0) are equal to set A, and which are not equal to it? Explain how to write it down. A A A B C D = ≠ ≠
Slide 7
How many elements does it contain:
Many days of the week? Lots of desks in the front row? Many letters of the Russian alphabet? Does Murka the cat have many tails? Does Petya have many noses? Lots of horses grazing on the moon? If a set has no elements, then it is said to be empty. The empty set is denoted as follows:Ø. Come up with some examples of an empty set.
Slide 8
http://www.kids-price.ru/kurnosiki_nabor_igrushek_dlya_vannoj_689446.html http://www.chicco-land.ru/product_info.php?products_id=231 http://www.serejik.ru/shop/good_460 http:/ /www.map.qcd.ru/igrushka-sobaka http://www.softtoys.com.ua/component/page,shop.browse/category_id,77/option,com_virtuemart/Itemid,38/ http://www. 56047.ru/shop/index.php?productID=3090 http://www.teddy-toys.ru/elephant http://www.elephant.ru/index.php?firm=160&type=106 Assignments from the textbook Mathematics 3rd grade ., author. Peterson L.G., M: Balass, 2010. Materials used: Author of the presentation, primary school teacher, Municipal Educational Institution Secondary School No. 9, Safonova, Smolensk Region, Irina Nikolaevna Korovina
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