129 - מושגים בסיסיים בתורת הקבוצות

Summary

In this video by TECHNION TEACHES, Dr. Aliza Malek introduces basic concepts in set theory as it relates to vector spaces and group theory. The lesson revisits the concepts of vector subspaces and introduces key set theory notations and operations such as intersection, union, and difference. Using practical examples and visual aids like Venn diagrams, Dr. Malek illustrates how these operations work and applies them to mathematical groups, setting the foundation for further studies in vector subspace theory.

Highlights

• Dr. Aliza Malek revisits vector subspaces to build upon set theory in group contexts. 📚
• Key set theory operations covered include intersection, union, and difference. ➕➖
• Using Venn diagrams, the lesson visually explains the relationships between groups. 🎨
• Real-world examples, like Messi's relationship to Barcelona, make the concepts relatable. ⚽
• Understanding that a group can be a subgroup or contain empty sets aids in grasping complex ideas. 🎓

Key Takeaways

• Set theory fundamentals are crucial for understanding vector spaces! 🌟
• Operations like intersection, union, and difference help define relationships within groups. 🧩
• Visual aids like Venn diagrams make learning these concepts easier and more engaging! 🎨
• Understanding these basics prepares students to delve deeper into vector subspaces. 📚
• Each concept builds on foundational math principles, tying back to everyday analogies. 🤔

Overview

In this engaging lesson, Dr. Aliza Malek from TECHNION TEACHES takes us through set theory fundamentals, linking them to vector spaces and groups in mathematics. The video kicks off with a brief recap of vector subspaces, refreshing our memories before diving into set theory.

The lesson emphasizes understanding operations like intersections, unions, and differences between groups, crucial for mathematical reasoning. Dr. Malek makes clever use of Venn diagrams to visually demonstrate these concepts, ensuring comprehension through visuals.

Relatable analogies, such as how Messi 'belongs' to Barcelona but not vice versa, ground complex terms in familiar contexts, helping solidify understanding. These foundational concepts in set theory pave the way for advanced study in vector subspaces, making the video a vital stepping stone for aspiring mathematicians.

Chapters

• 00:00 - 00:30: Chapter 4 - Vector Spaces Chapter 4 introduces the fundamental concepts of vector spaces, building on the principles of linear algebra. Dr. Aliza Malek reviews definitions of a vector space and a vector subspace, providing numerous examples to illustrate these concepts. The chapter aims to develop further theories in vector spaces and begins by revisiting foundational concepts in group theory to enhance understanding. Key terms include vector space, vector subspace, and group theory.
• 00:30 - 01:30: Vector Subspaces and Group Concepts The chapter discusses vector subspaces of a vector space V, explaining that W is a vector subspace if it is also a vector space with the same operations and over the same field. It highlights that subspaces are actually subgroups of a larger space, linking the concept to group theory. The chapter then transitions to recalling concepts related to groups, including operations and notation.
• 01:30 - 04:30: Group Belonging and Containment The chapter titled 'Group Belonging and Containment' introduces familiar mathematical notations used to represent the concept of belonging within a group. A particular notation, resembling the letter 'E,' signifies that an item belongs to a group. For instance, numbers like 1 and 3 are shown to belong to the group of natural numbers, denoted as N, indicating they are natural numbers.
• 04:30 - 09:30: Operations Between Groups The chapter discusses the concept of group belonging in mathematics, using the example of numbers and sets. It explains that -3 does not belong to the set of natural numbers (N), as indicated by the 'does not belong' symbol. The chapter further clarifies that a group cannot belong to an individual number, using an analogy that Messi belongs to Barcelona to illustrate the point.
• 09:30 - 18:30: Group Sum and Examples The chapter discusses the concept of group membership and notation. It clarifies that terms belong to a group rather than groups belonging to terms. The chapter introduces a new notation signifying 'contained in,' which is used to express that a group is contained within another group. The discussion uses metaphoric examples, such as Barcelona and Messi, to illustrate the point.
• 18:30 - 28:00: Official Definitions and Features of Groups This chapter introduces the concept of a group using the example of terms belonging to a group. It explains that groups are indicated with curly brackets and shows how individual terms such as 1 and 3 can be grouped together using these brackets. The chapter identifies groups as collections that contain specific terms, demonstrating the idea by grouping 1 and 3 within curly brackets.
• 28:00 - 31:30: Vector Space and Vector Subspace Insights The chapter explores the concepts of vector spaces and vector subspaces, with an emphasis on understanding the relationships between different sets of numbers. It discusses how certain groups, like \(\{1,3\}\), can be subsets of the natural numbers, while clarifying that other sets, like the integers (denoted \(\mathbb{Z}\)), are not contained within the natural numbers. However, the natural numbers are subsets of the integers, illustrating a clear hierarchy and relationship between these sets. The discussion may also touch on the representation and notation of these mathematical structures.
• 31:30 - 33:00: Conclusion and Introduction to Further Theory Development The chapter discusses the concept of an empty group in mathematical terms. It explains that the empty group can be represented by the letter phi or empty curly brackets. The analogy of an empty bag is used to describe the empty group as having no terms. Furthermore, it's noted that the empty group is contained within every other group, symbolized as group A, regardless of its composition.

129 - מושגים בסיסיים בתורת הקבוצות Transcription

• 00:00 - 00:30 Chapter 4 - Vector Spaces Basic Concepts in Group Theory Linear algebra course Dr. Aliza Malek Hello, thank you for coming back In the last lessons we saw what a vector space is, what a vector subspace is, and we have seen a great many examples of vector subspaces. Now we would like to develop theories. For this, let's recall some basic concepts in group theory. Let's get going. First we recall again the definition of a vector subspace, I have V, a vector space over F, we have W, a subgroup of V,
• 00:30 - 01:00 W is called a vector subspace of V, with W a vector space in itself, with the same operations, and over the same field. Because subspaces are actually subgroups of a larger space, we want to speak in terms of groups. And so we will now recall the concepts related to groups. So we will recall the different operations and notation that we know.
• 01:00 - 01:30 So here are some familiar notations that we may have seen before, but we will list them anyway. This sign, sort of an E, this way, it is a sign that represents belonging. A term that belongs to a group, for example, 1 and 3 belong to N, N is the natural numbers. 1 is natural, 3 is natural, so 1 and 3 belong to the natural group N.
• 01:30 - 02:00 On the other hand, -3 does not belong to N. When we write the "belongs to" sign and strike through it, we mean it does not belong, because -3 is not a natural number. Can we also say that N belongs to 3? Does the group belong to 3? Definitely not. To me it reminds me of something very simple: Messi belongs to Barcelona.
• 02:00 - 02:30 Barcelona does not belong to Messi. Or maybe yes? This is a topic for discussion elsewhere, not here. But I hope that you got the idea. So we don't say that the group belongs to the term, rather, the term belongs to the group. Here is another notation. It is very similar to the previous notation but does not have that line in the middle. This is a sign of "contained in". When we talk about groups, a group is contained within another group,
• 02:30 - 03:00 a term belongs, a group is contained. We indicate that one group is contained in another group. Let's see an example. The group, the curly brackets indicate a group, pay attention. Here 1 and 3 are single terms, so they "belong", on the other hand, here we put them in a group. The curly brackets put 1 and 3 in a group. This is a group that contains 1 and 3.
• 03:00 - 03:30 So the group 1,3 is contained within the naturals. 1 and 3 belong, the group with terms 1,3 is contained in the naturals. For example, you can say that Z, the integers, is not contained in the naturals. The naturals are contained in the integers, but the opposite will no longer be true. The sign strike through 0,
• 03:30 - 04:00 the letter phi, or completely empty curly brackets with nothing inside, mark the empty group. The empty group is simply a group without terms. You can think of it like an empty bag with nothing in it, this is an empty group. It is worth noting that the empty group is contained in every other A group, no matter where, that is, the empty group is contained in A.
• 04:00 - 04:30 An empty group is a group, therefore the sign of containment is used, not belonging. The empty group is contained in A for every group A, no matter which group you take, the empty group is contained within it. Here are operations between groups we know, and let's recall three main operations which you have probably seen before. We are doing it now in simple language and in a simple way.
• 04:30 - 05:00 Soon we will make everything official. So the first operation is an Intersection. What is an Intersection? The collection of all the terms shared by the groups. This is what we called an Intersection. How do we mark an intersection? With this sign. Reminds the letter U upside down. That's how you can remember. Union: the collection of all terms that are in at least one group. How do we mark this?
• 05:00 - 05:30 Very similar to what is here, but flipped. Now think about it. The word Union starts with a U. well, it is a kind of convenient way to remember Union. We have Difference. This Difference is a collection of all the terms that are in one group, but not in the other group. How do we mark a Difference? A Difference is marked either with a slash, or with a minus. I will prefer the slash, you will soon understand why.
• 05:30 - 06:00 So let's recall once again. An Intersection is an upside down U. A Union is a regular U. And the Difference will be either a slash, or a minus. These are the signs we will use. Now we will introduce another operation that you probably have not encountered before. A Sum, what will be the sum of two groups?
• 06:00 - 06:30 It is a collection of all the terms that we get by adding them up, a term for the first group to a term from the second group, just add them up. Of course this means that we must know how to add them. If we cannot add the terms from group A to group B, we will not be able to define the sum group. An addition must be defined between these terms. For example, we shall recall that I have numbers and I have matrices,
• 06:30 - 07:00 I don't know how to add numbers and matrices, so I won't be able to add a group of numbers with a group of matrices. I don't know how to add matrices of a certain rank with matrices of another rank. No addition will be defined there. But if I have a group of matrices with a group of matrices of the same rank that I know how to add, I can talk about the sum group. We will see examples of this shortly. The notation will be, a plus.
• 07:00 - 07:30 A Difference does not mean to subtract. A Sum means to add. So it's a bit confusing. So I would prefer to use a slash for the Difference. Don't think for a moment that Difference between groups means a difference between terms. Although Sum between groups means a sum, a sum of terms. It's a bit confusing, so be careful. Let's see an example.
• 07:30 - 08:00 Here are two groups, A contains the numbers 1,2,3,4,5, and B which contains the numbers 2,4,6,8. Here, we will not prove things in an official way. That's why we will use a drawing, here. This is what we call a Venn diagram. This drawing represents group A, and the other drawing, the brighter one, the other blue color, represents group B. We will use drawings to see the concepts we are talking about,
• 08:00 - 08:30 we will not prove things from a formal mathematical point of view. What will be A intersecting B? It is the terms shared by A and B. Which terms are in both A and B? If you look at these groups, it is very easy to see that the shared terms are 2 and 4. Where do I see it in the drawing, here: 2 and 4 are in the shared area between A and B, in both A and B.
• 08:30 - 09:00 A Union B. These are the terms that are in at least one group. Either in A, or in B. And note that we write each term only once. 2 and 4 are in both, though I don't write them twice. I write them only once, here is 2 once, and here is 4 once. Soon we will place A Union B in the drawing, in the meantime let's see its area, where is it situated, here: The red area that surrounds the two groups is the Union.
• 09:00 - 09:30 Now we take A, the difference between A and B, A minus B. We take A, and we gently discard the terms that are in B. If you take A, and discard 2,4,6 and 8, we removed 2, we removed 4, 6 and 8 don't need to be removed because they weren't there anyway, and so we remain with the terms 1,3,5.
• 09:30 - 10:00 Where are they in our drawing, here: We actually take group A, and we need to remove the terms that are in both A and B. Because those that are in B and not in A, do not need to be removed in advance. And so we take A, and discard A intersecting B, 2 and 4. Now notice that B minus A, the difference between B and A,
• 10:00 - 10:30 will not be the same as between A and B. Because now I will take B from B, and from it I will remove the terms of A, or, in fact, the terms shared by A and B. I'm going to take B, and I'm going to remove the shared terms 2 and 4. What will I be left with? Right, with 6 and 8. Where are they located in the drawing? Over here. And now when I look, I see the Union. 1,3,5,2,4,6,8 is the Union.
• 10:30 - 11:00 It is divided into terms of A minus B, A intersecting B, and B minus A. This is how the terms of A Union B are dispersed. You can see that these are all the terms that are listed here. What will be A plus B? We can add numbers of A and B, because they are natural numbers. We will use addition of numbers as we know it. So what will A plus B be? We need to take a term from A, and a term from B, and add them.
• 11:00 - 11:30 A term from A, and a term from B, and add them together. If we do it in an orderly fashion, what do we get? We take 1, and we need to add it to 2,4,6,8. Which terms do we get? 3,5,7,9. 3 ,5, 7, 9. Now we will take 2, and add it to 2,4,6,8. Pay attention, terms we have already written, we will not write twice. So if we add 2 to 2,4,6,8,
• 11:30 - 12:00 we will get 4, 6, 8, 10. 4, 6, 8, 10, and so on, until we get all the terms, one from here with another from here. We will get a large group. Note that in A plus B, sometimes, not always, we have terms from A, for example 3, terms from the intersection, for example 4, terms from B, for example 8. But there are also terms that were neither in A nor in B.
• 12:00 - 12:30 For example, 11,12,13. These are completely new terms that were created from the addition of the two groups. It is also worth noting that the sum group has terms that can be obtained in only one way, 13, for example. I only have one way to get 13. Take 5 from A, plus 8 from B. That's it, it is the only way we can get 13. On the other hand, if I now look at other terms, such as 7,
• 12:30 - 13:00 it can be gotten in many ways, let's see. I can get 7 as 1 plus 6. 1 from A, plus 6 from B, gives a 7. Wait, we can also take 3 from A plus 4 from B, this will also give 7. And I even have another way to do it. I can take 5 plus 2, another way to do it. Of course, we will only write 7 once, but I have many ways to get it.
• 13:00 - 13:30 Now let's go to the official definitions of the groups. I have A and B, two groups. We will define the following operations: An Intersection between groups. The notation A intersecting B gives me the new group, which equals to all x terms, such that x belongs to A,
• 13:30 - 14:00 and x also belongs to B. x is in both A and B. We will move to Union. A union B, remember the notation, A union B is equal to all x terms, such that x belongs to A, or x belongs to B. It is enough to be in just one of the groups. A Difference between groups. A minus B, or A/B is equal to all x terms, such that x is A, and also that x is not in B.
• 14:00 - 14:30 And if we want B minus A, then of course it's all the x terms that are in B, and also are not in A, do not belong to A. And if an addition is defined between the two groups, then we can also define an addition. We will define a sum between the groups by A plus B is equal to all terms of the form x plus y, when
• 14:30 - 15:00 x belongs to A, and also y belongs to B. We need to make sure we have terms that can be added together. Now, we would like to recall the features. Of course we will not go into proving here, I will just show you this inside our drawing. If you would like to, you can try to write things in a formal way but what is important is that you remember the features
• 15:00 - 15:30 because we will use them during this course, and in the following discussions we will have, in vector spaces and vector subspaces, which are all groups. So here is our drawing, here is group A, here is group B. And let's see the features we would like to talk about. A intersecting B is the same as B intersecting A. Indeed, it does not matter to this shared area which we mentioned first, A intersecting B, or B intersecting A.
• 15:30 - 16:00 Same goes to the Union, in the end we get the same thing whether we do A union B, or B union A. But A minus B, and B minus A will no longer be the same thing. A minus B will be this area over here: And B minus A will be the area I have left over here: It's not the same thing anymore. Not every operation between groups has a commutative law, not every operation is commutative. A intersecting B is contained in both A and B.
• 16:00 - 16:30 Yes, A intersecting B is this blue area, which sits inside A and also sits inside B. It is contained, the intersection is a group, the group is contained in both A and B. Conversely, both groups A and B are contained in A union B. A union B, as you remember, is the two groups together, A sits inside, and B sits inside.
• 16:30 - 17:00 Therefore A is contained in the union, and B is contained in the union. That we have already said... That the empty group is contained within A, it is contained in every group, but A is also contained in A. Think about it, all terms of A belong to A, therefore, A is contained within A. It seems a very simple thing, but it is important to remember it.
• 17:00 - 17:30 Each group is a subgroup of itself, it is self-contained. The empty group is also contained in every group. A minus B is contained within A. If I take A, and remove the terms of B, what remains is obviously a subgroup of A, contained in A. A minus A is the empty group, right? Take A,
• 17:30 - 18:00 delete, throw away, take out all the terms of A, what will you be left with? With nothing. What does it mean to be left with nothing? It just that, being left with the empty group. Nine, if by chance, if I have some defined addition in A and B, and this addition has a commutative law, that is, x plus y is equal to y plus x, then it would also be possible to say that A plus B is equal to B plus A.
• 18:00 - 18:30 If I know that you taking a term from A and adding a term from B to it, and that this addition has a commutative law, like adding numbers, like adding matrices, then we can say that A plus B, or B plus A, ends up being the same. That's it, these are the features we would like to remember, that are related to operations between groups. A question: To what will the operations be equal to if A is contained in B?
• 18:30 - 19:00 A is a subgroup of B. What happens then, let's see. First we will see a drawing. Here is B., and A is contained within B. Here A is sitting inside B. A is part of B. Why is it important to us? Remember, We will have a vector space, and we will have a vector subspace. That is why it is important for us to check what happens in such a case. So let's see what comes out.
• 19:00 - 19:30 A intersecting B will be A, because these are the terms shared by B and by A, of course it's A, it's just the small group. If group A is contained in group B, the intersection is the smaller group. Why is A union B equal to? Of course, when we take the A union B, every term that is in A, or in B either way is in B, and therefore the Union will be group B.
• 19:30 - 20:00 And what will be the Difference between A and B? Take A, and take the terms of B out of it. Since every term of A is within B, we removed everything, and what are we left with? with the empty group. Our mascot asks what happens when we do B minus A? An interesting question. So here is B, let's gently discard the terms of A, we are left with a doughnut. Indeed, it is delicious. this is a doughnut.
• 20:00 - 20:30 We took B, made a hole, we are left with a doughnut, our mascot is right. That's it guys, we are done for now. Now that we are familiar with the basic concepts of the group theory, we can get started, and develop the theory within the vector subspaces. Thank you very much.