Aula de Matemática - Supletivo 2026

追加: 2026-05-12

[00:00:00]Hey guys. Goodnight. How are you all doing?

[00:00:14]Today we'll have our math class, but I have a little message for you all.

[00:00:44]Okay guys, come on, I'll wait a little bit, okay? So we can start our class, those 5 minutes until everyone has had time to log in. Let's grab some paper and pen right now, okay? These are what you'll need today to take notes. Actually, you always need it, right? But mathematics is much more so.

[00:01:04]You know that math is all about practice, right?

[00:01:10]So you need to practice.

[00:01:13]And I also have a little message for you all. I'll send you a link through the chat that Professor Daniel requested, okay?

[00:01:23]I'm just waiting for it to open here and I'll forward it to you.

[00:01:36]Hey everyone, it's an attendance list, okay? You need to sign this, OK?

[00:01:43]So, starting with today's class, you need to sign this attendance list.

[00:01:48]So, you're going to click on the link that I'm going to send you, OK?

[00:01:58]So, just wait, I'll send you the link, okay? Ah, I'm just waiting for it to load here so I can send it to you.

[00:02:10]Meanwhile, I'll post our lesson for today here.

[00:02:27]Today's lesson is on page 115.

[00:02:37]I'll leave it here in the chat, okay? So, those students who arrive late will have to check the chat, okay?

[00:02:50]As for today's page. Mathematics, page 115. Okay. Okay everyone, let's begin today's lesson.

[00:05:57]No, guys, nothing froze, I just stayed quiet. Okay, everything's fine here.

[00:06:09]Just a moment, folks, things are about to change because I'm switching networks here since it switched to 2G. I've been doing well, everyone, so what are we going to see today? Let's begin our lesson now. Stay until the end, okay everyone?

[00:06:37]in order to sign the attendance list.

[00:06:45]Okay, so today we're going to look at number sets, right? So you're going to access your workbook, page 115. We're going to look at these number sets, okay? What does that mean? In mathematics, when we talk about sets, what do we think of? ( The numerical sets, folks, in mathematics—well, not just in mathematics, but when we talk about sets, what do we think of?) Several similar things put together, right?

[00:07:16]So, a set can be a collection of objects, a collection of pens, for example, right? A set, a collection of elements that will have some characteristics in common, okay? And then we can use sets, for example, to make a list, right? For example, a list of people, a shopping list, right?

[00:07:46]And here we're going to see how these sets are classified within mathematics. So, we will have the natural sets.

[00:07:57]So, here we have the numbers that are called natural numbers, which are the numbers that we use to count things, to order things. So, the set of natural numbers is represented by the letter N, okay? And whenever I have a set, guys, we're going to use those keys, okay? So, you see that n = key? Open the key. So, all the natural numbers will be understood to be the numbers 0, 1, 2, 3, 4, 5, 6, 7, and so on, to infinity, okay? So, for example, there are five chairs, right? There are not five chairs in the room. So, I counted how many chairs there are in a room, okay? So this number five belongs to a set of natural numbers, right? Another example, the runner, right, came in first place. So, in first place, right? There, we're also ranking them. First, second, third, it enters into numbers, right, natural numbers. Okay, everyone? There was just a slight formatting issue there because of when I switched to PDF, okay? But those are the sets of integers, right? So it's represented by the letter Z.

[00:09:29]These sets, okay? We're going to include all the positive numbers, that is, those numbers greater than zero, the number zero, and here the numbers less than zero, which are those negative numbers, right? We've already covered negative numbers a little bit in previous lessons. Therefore, these numbers belong to the set of integers.

[00:09:55]For example, when we use negative numbers, -5º, right?

[00:10:02]What are they? These are numbers, right, where we have a negative temperature, a temperature below zero, right? Your bank balance is R$100 less. So, when it's in the red, you know, that negative account we talk about, right? When you owe money to the bank, then all those numbers fall into the set of integers. OK, guys? I'm leaving now, I'll let you know again. Whenever you guys are late, I'll always leave the page number and subject of today's class in the chat, okay? So, when you log in, check the chat to see if there's a message there with the page number, OK?

[00:10:49]And that's today's topic, okay? The next set we're going to look at now is the set of numbers that we call rational numbers.

[00:11:02]What numbers are these? So, those are the numbers that are written as fractions, right? So, a fraction has a numerator and a denominator. Numerator on top, denominator on the bottom, right? So, when we were there, I gave the example of the pizza, right? The pizza has eight slices, right? If you eat two, then it 's 2 for the pizza. Or if it's a small pizza, four slices, you ate one slice, 1/4, okay? So these numbers are written as fractions. Other numbers that come into play, okay? Also, rational numbers are decimal numbers, that is, those numbers that have decimal points, right?

[00:11:49]And decimal numbers that contain commas, right?

[00:11:55]OK. The next numbers we're going to look at are the set of irrational numbers.

[00:12:02]Well, these numbers are decimal numbers, meaning they will have decimal points, but they have a characteristic that we call that they are not classified as a repeating decimal. A repeating decimal is how we have these repeated numbers, you see? At least two repeated numbers. So, when, you know when we do a calculation on a calculator, we divide one number by another and it will give us 0, and then it keeps repeating the numbers, okay?

[00:12:37]So, we consider it a repeating decimal when we have two identical repetitions. This one has at least three, okay? which will not happen in the case of irrational numbers.

[00:12:49]Irrational numbers will also have decimals.

[00:12:52]So, in the case of pi, the number pi is 3.149. It has several digits after the decimal point, but they are not repeating numbers, so they are considered irrational numbers, okay?

[00:13:11]And here it was the square root of 2, okay? The little key we use didn't come out here, okay? When it's time to do the conversion. So, we have, look, 1.41 42 1, okay? And these numbers can also be transformed into fractions, they cannot be transformed into fractions. So, if they could be transformed into fractions, they would be rational numbers and not irrational numbers. Is everything alright here, everyone?

[00:13:51]This first part.

[00:13:54]Perfect.

[00:13:55]And now we're going to look at the set of real numbers. which will be represented by the letter R, okay? What does the set of real numbers mean? It will encompass all the numbers. So, we're going to have rational and irrational people.

[00:14:15]So, these are all the numbers that we can fit into a straight line, okay?

[00:14:24]So, reviewing here, to help you understand the differences, natural numbers, represented by the letter N, are positive integers or non-negative integers. So we'll have 0 1 2 3 4 and so on.

[00:14:44]The integers are represented by the letter Z, and this set includes positive numbers, negative numbers, and also zero.

[00:15:01]Rational numbers are represented by the letter Q, and these are the numbers from which we will find fractions, right?

[00:15:10]And what about repeating decimals, right?

[00:15:14]And the integers also fall into that set Q. And the set of irrational numbers will be represented by the letter I. And here we'll have non-repeating decimals, okay? And also inexact roots, for example, the square root of 2, okay?

[00:15:35]And within the groups of real numbers, we 'll have all the other numbers.

[00:15:41]Sorry everyone, I ended up skipping over all the other numbers, right? The ones that fall under both rational and irrational numbers, meaning it encompasses all numbers, correct?

[00:15:57]And here, folks, we're going to understand how they relate to each other, okay? So, if we were to create a way to arrange these sets, we would see that we have natural numbers here, right? Z encompasses both Z and natural. The rational numbers include, right, the Z numbers, the integers, and the natural numbers.

[00:16:28]The real numbers will encompass, right, irrational numbers, rational numbers, integers, and natural numbers. And the irrational beings, you notice that they are kept separate from the rational beings, rational and irrational, separate, right? And reality ends up encompassing all of them, right?

[00:16:50]All good here, folks, for this first part.

[00:17:08]Guys, have you received the attendance list to sign yet?

[00:17:12]I'll send the link here, you guys come here and see if you can sign up, okay? I'm just waiting for it to download because I have to access the link from my computer and it's taking a long time to load, okay? That's why I asked you to wait there, probably until the end of class, so I can send you that link in the chat, okay?

[00:17:34]Okay, let's move on. And now we're going to look at what are called real intervals. So, we have to think about the numbers within a straight line, right?

[00:17:44]So, we have the line 0 1 2 3 4 5, which are the positive numbers, right? And beyond zero, which are the negative numbers, 1, -1, -2, -3, -4, -5. We had already seen this in previous classes. However, have you ever stopped to think that between the number zero and the number one there are many numbers?

[00:18:10]So, what we'll have as a subset of these real numbers are those numbers that are in those intervals that we don't normally count. Have you ever stopped to pick up a ruler? Let me see if I have a ruler here.

[00:18:31]A ruler, okay? I have a ruler here. Using a ruler, we'll have centimeters here, right? So, from 0 to 1 I have 1 cm, from 1 to 2 I have 1 cm, and from 2 to 3 I have another 1 cm. And have you noticed that there are several dashes from zero to, or from one to two, or from three?

[00:18:56]What this means is more or less what I'm showing you here, which is an interval, that which we have between one number and another. In the case of the ruler, since it's in centimeters, those little lines represent millimeters, okay? Which is the other unit of measurement.

[00:19:16]So that little line that comes after the zero, 1 mm, 2 mm, 3 mm, until we have 10 mm. 10 mm is equal to 1 cm, okay? So here, for example, in this image that I put on the screen for you, you are seeing a straight line, a straight line, where we have several numbers and we will have several other numbers within this interval. For example, from -5 to -4, we'll have a series of numbers.

[00:19:47]From -4 to -3, another series of numbers.

[00:19:51]from -3 to -2 plus a series of numbers and so on. So these are the intervals, okay?

[00:20:01]And then we'll write those intervals. And so you can see that there's an oval shape above that line that encompasses the numbers from zero to five. So when we write this as a set, we'll write it in the following way shown below. Do you see that it has the letter A? The letter A would represent set A, it's represented like that, right?

[00:20:31]Hypothetically, the set A equals the key X E. See that E? It means " belongs to," it's the symbol for "belongs to" in mathematics. Then he says that X belongs to the real numbers, such that then that bar in the middle such that 0 is less than or equal to x which is less than or equal to 5. Close the brace. Okay? What does this represent? This set, where this oval representation is located on this line. So it encompasses all the numbers within this range from zero to c.

[00:21:18]So he's being represented in this way, okay?

[00:21:26]Hey everyone, we need to remember a few things. Right? Some things you learned back in elementary school, like the greater-than sign and the less-than sign. Do you remember this?

[00:21:45]So, I put the representation there for you. The bigger sign, okay? He's going to represent, you know, express that the number in front of the figure is greater than the one behind it. So he's going to have that symbol that's over there, okay? And remember, here it will always be in the position with the higher number, okay? It goes in front of the smaller one, right? So there, I have 3 greater than 2.

[00:22:30]3 is greater than two, isn't it? Okay?

[00:22:33]So the symbol will always be the larger one, the symbol of the larger number, and the smaller number, okay? And the youngest one? The smaller one is the opposite, okay? For the smaller number, we'll have the smaller number in front, the less-than sign, and the larger number after, okay? So this will indicate that the first number is smaller than the larger one. So we're going to have 2 less than 6, okay? There are a few tricks we can use to identify this.

[00:23:09]As I showed you in the little image, you've probably heard of the alligator's mouth, right? So, the alligator's mouth will always be facing the larger number. Look at the cake! So, we have the small piece of cake here, okay? And the alligator's little mouth is turned towards the cake, which is bigger. So this cake is smaller than that other cake, okay? So look at the difference from the one below. The one on the bottom has a large slice of pizza. The alligator's mouth will always be turned towards the biggest one, okay? So the alligator's little mouth will eat the bigger piece as well as the smaller piece, okay? So here's another way to make this easier for you, because sometimes the greater than and less than signs can be confusing. So it's very simple, you just have to do it, take the signal and make a cut. Do you see that second image where the letter 'o' is and where it says 'four the smallest'? So, if you take it and draw a line, it becomes a four.

[00:24:25]If it turns into a four, it is smaller.

[00:24:30]The figure on the side, right? Next, it is the symbol of the greatest. If you draw a line, it turns into a seven. So, seven is greater. So, this is a way for you to more easily identify whether a sign is greater or lesser, right?

[00:24:57]Well, Brenda, not always. The biggest one, yes, the biggest one will always be the biggest one, the alligator's mouth, remember? She will always be facing the number. We need to see what those two numbers are that are being represented, right? For example, I'll go back to this slide so you understand. For example, here I have three greater than two.

[00:25:25]So, if three is greater than two, the sign will have to face the three. Remember that the alligator's mouth will always eat the biggest one, okay?

[00:25:36]And when it's the smaller sign, it's the opposite.

[00:25:39]So, we'll have the smaller number in front, the less-than sign, and the larger number after. That's the correct position, okay? Okay everyone, another way we're going to use a lot now in this section on sets is the symbol called "greater than" or "equal to," okay?

[00:26:06]The greater than and equal to symbols. We're going to have the symbol of the larger one, which is that little alligator mouth, but it's going to have a line underneath. Remember the equal sign? The equal sign is represented by two dashes.

[00:26:17]I'll put it there for you in the chat.

[00:26:20]Equal sign, right? The equal sign is represented by two dashes.

[00:26:25]Now, when I want to say that something is greater than or equal to, I'm going to use the crocodile's mouth and put a line, a line of the equal sign. And we're going to do the same thing for the younger one, okay? The least or equal. Smaller is the same as the crocodile's upside-down mouth, right? Always there for the bigger one, okay? So we'll have the smaller number in front and the equals sign, right? A small equal line will appear underneath, as shown in the slide. So, we have an example there, x greater than or equal to 5, x = 4, right?

[00:27:14]So, to represent these sets, we can use a straight line representation, okay? So when we represent these sets, we're going to use these little circles in those intervals. So, the filled circle will represent that a number is part of the interval, okay? We say "closed" and "the little circle is open," meaning the little circle is not filled in, okay? She's indicating that that number doesn't belong to that group, it doesn't belong to that range, okay? And it is called open. We're going to see that in the next few slides, okay? So let's go. So now we're going to look at the types of intervals. Regarding interval types, folks, we're going to have what we call a closed interval, okay? This closed interval will include all the numbers on the line.

[00:28:21]So we can write this interval in notation, which will be using parentheses. So, do you see A and B over there?

[00:28:32]So, let's say A and B are part of a set of numbers, okay? So, if they are part of this set, we 'll put the parentheses in closing, both on side A and on side B. And then I can represent this interval in set form. So, remember that we're going to use the key x belongs to the real numbers, such that a is less than or equal to x men.

[00:29:02]Turn off the key, okay? So, remember that in this case, we'll always use the symbol ☐, the less-than sign, okay? Here, since it's a closed interval, we'll use the less than or equal to symbol. It will be easier for you to understand now in this next image, okay, everyone? Just a little bit, I'm waiting for the link to load. He's at 76%.

[00:29:32]It's because I have to open my WhatsApp here to get the link, and that's taking a while, okay? Let's understand, then, how we're going to represent a closed interval. So, let's say I have it there, look.

[00:29:47]So, we're going to have the bracket facing inwards here. What did I say? Parentheses, right? No, it's not parentheses, it's brackets. Sorry, but we can use parentheses as well.

[00:30:00]bracket. A square bracket facing inwards indicates that the number belongs to that group. He's part of that group, okay?

[00:30:09]Another important point is that the less than or equal to sign will tell you that it belongs to this group, okay? So we're going to put, like, A through B. And then we're going to make this little worm here, this little snake here on top of the line, okay? This indicates that I have a closed interval. Therefore, a belongs to the set.

[00:30:33]They both belong to the same set, okay? Here's another representation, the number three and the number eight. So I have a line where I have an interval that goes from three to eight. So I want to say yes, the number three and the number eight, they belong to the same set. So the little ball is filled in, that's why the little ball there is filled in pink, okay? And I'm going to draw this little snake on top of the line, saying that all the numbers between three and eight belong to that same set. So I can put this notation in the form, right, in the form of a set that will be X belongs to R, to the real numbers, such that 3 min ig x less than or equal to 8, okay?

[00:31:30]Now we're going to see the open interval.

[00:31:33]So the open interval is the opposite, okay? So here, as I said, we can... the open interval indicates that the numbers are not part of this set, okay? So what exactly is it? The numbers that are at the extreme end of a given number to that number. So how is this going to work out?

[00:31:57]If we use annotation, remember here, the bracket will be on the outside, unlike when it's closed. When closed, the clasp faces inward.

[00:32:11]So, if I put a bracket facing outwards, that will tell me that A will not be included within this interval, nor B, only what is between A and B. And if I were to write this in set form, it would be X belongs to the real groups, such that a less than X is less than B. So what is the difference between this sign and the other?

[00:32:42]This one here we had a smaller equal one and this one here the symbol is smaller.

[00:32:51]Example. So we have an interval here between 2 and 5. So this interval, ah, and observe here, okay everyone? When we use parentheses, the parentheses go inside the parenthesis and it will start right after the colon.

[00:33:07]So, if I have an interval on a number line that represents the numbers from two to five, I'm not going to include the number two, nor the number five. So, what are the numbers between two and five?

[00:33:21]3 and 4. So, this set will only represent the numbers 3 and 4.

[00:33:28]Two is not included, and neither is five.

[00:33:32]Let's look at it in a simpler way so you can understand.

[00:33:39]Let me just check something here, guys. That's wrong. Oh no, no, that's right. This is a bracket. So, with brackets we have the closed position. And here the difference is the open parenthesis, okay? But we usually use this one more.

[00:34:04]So, let's go. Oh, bracket facing outwards, parentheses facing inwards.

[00:34:09]This will tell me, it will indicate to me that I have an open interval, okay? Another characteristic of an open interval is that we'll have the "less than" sign, not " less than or equal to," okay? Another characteristic is that the ball is empty between the intervals. So, let's look at this representation of this line. So there I have the representation of 3.8.

[00:34:38]Notice that the brackets are facing outwards, so this indicates to me that the circle is empty. So I'm going to represent the number three and the number eight on the number line. Empty ball in both.

[00:34:52]So what does that mean? that only the numbers between 3 and 8 belong to this set. What will the numbers be? 4, 5, 6, and 7. The numbers three and eight are not included in this set, okay? If we were to represent the set, it would be X belongs to R, such that 3 is less than X, which is less than 8, which is what I just said. What are the numbers between 3 and 8? 4, 5, 6 and 7.

[00:35:28]OK? That's it, folks.

[00:35:37]Next. Now we are going to look at the interval that is called semi-open or semi-closed. What does this indicate?

[00:35:45]We're going to include the numbers on only one end.

[00:35:50]So, we're going to have one empty ball and one filled ball on this straight line, okay? So here, for example, in this example I gave, we have a bracket, right? And the letter A. So this indicates that A is included in this set, but we have a B in parentheses. This indicates that he is not included, okay? So, how are we going to write this out as a whole? X belongs to the set of real numbers such that A is less than or equal to X, which is less than B.

[00:37:03]Right, everyone.

[00:37:05]Guys, I'm just waiting here, it's at 89%, I'll send you the link soon, okay? So you can sign the attendance list. Let's take a look at this representation here.

[00:37:16]See what I said? So when we have this situation, we're going to have an open ball, a closed ball, a closed ball, an open ball. So here I have a representation. A bracket facing inwards indicates that the ball is inflated. The number belongs to the set.

[00:37:35]When I have a bracket facing outwards, it indicates that the number does not belong to the set, so the circle will be empty. So I'm going to represent that with a straight line: 3 and 8. The number three belongs, which is why it has a filled circle, and the number eight doesn't belong, which is why it has an empty circle. If I were to represent this in set form, it would be X belongs to the real numbers, such that 3 less than or equal to. Remember that we always use the less than or equal to symbol when we have the shaded circle, the X, which means less than eight. Why did I use "less than not less than equal to"? Because the ball is empty. So always associate the less than or equal to sign with a filled circle.

[00:38:23]Sign of a smaller empty circle. So, what numbers correspond to that range? 3, 4, 5, 6, and 7. But not eight, because eight has an empty circle. Is everything alright here, everyone? We're almost finished with this first part, okay, everyone? You received the attendance list in the group you're part of, okay? And I'll repost it here in the chat for those who didn't sign up there or who aren't part of the group and didn't receive it, okay?

[00:39:06]It's at 92%, it's almost finished. Please wait a few more minutes.

[00:39:12]Next.

[00:39:14]Next up, folks, are the intervals that we call infinity. Have you ever seen this symbol before? This symbol corresponds to infinity, so infinite numbers, okay? We can use this infinite number when we don't have an end to that line, okay?

[00:39:39]For example, I can use three. So, I put that example there for you. The closed bracket three plus, then indicates infinite positive numbers and the closed parenthesis. What does this indicate?

[00:39:59]It represents all numbers greater than or equal to three, okay?

[00:40:05]So, the infinity symbol will always be enclosed in open parentheses or brackets, okay? Why? Because it's a number that we don't know what it is. Remember the infinite number? Look at that image. It will be easier for you to understand.

[00:40:23]So, look, I have here an open bracket facing outwards, indicating open, minus the infinity symbol. So this indicates to me that it's a negative number. So, if it's a negative number, it's those numbers that are behind zero, right? In a straight line, since infinity is always open, it will not have any little balls at its end.

[00:40:51]But the number five has a closed bracket.

[00:40:54]This indicates to me that the ball is closed, a clasp turned inwards. So, closed circle. So, how will this stay in a straight line? I'm going to put the five, the closed circle, and I'm going to make the snake until the end of the line. And there won't be a number at the end, meaning there's no interval because the interval is infinite.

[00:41:16]And how will this written form look when used as a set? X belongs to the set of real numbers, such that X is greater than or equal to or less than or equal to, sorry, 5. So all numbers less than five will fall into this set, okay? The same thing will happen here in this side image. So I have three open, which indicates that the little ball is empty plus infinity. This plus sign indicates positive numbers, meaning it's moving forward, not backward. When it's a negative number, it's infinite; negative points go backward, and when it's infinite, it's positive; forward points go. So, since three is empty, the little circle will remain empty, and we'll fill in the entire snake line, right? It will fill in the snake and the entire straight line to infinity. So how will things turn out here? Here, folks, notice that it's going to be a little different.

[00:42:22]X belongs to the real numbers, such that X is greater than 3. Note that this was the only time we changed the sign, because before we only used the less-than sign. Now we're using a larger signal. Greater because x is greater than 3. Therefore, these are all the numbers that are greater than three. What numbers are greater than three?

[00:42:44]4 5 6 7 to infinity.

[00:42:49]OK?

[00:42:51]So, to recap, what's important here, this little square at the bottom, what do we have there? We have a parenthesis facing inwards, which indicates that it is open. The bracket facing outwards indicates that it's open, okay? So, whenever we have the brackets together, the parentheses together with the infinity symbol facing forward, or the bracket facing outward, we'll understand that it's an open bracket, okay?

[00:43:25]Which ones are most commonly used, teacher?

[00:43:28]Parentheses or brackets? The brackets, okay? I only put the parentheses here in case you noticed them.

[00:43:39]Now we're going to review set theory a little bit. You all must have studied sets at some point, right? So here we have set A and set B. These sets can be together. So you see that over there, right? A belongs to set B. In that part where A is inside the circle of B.

[00:44:03]So A, the set A, belongs to set B. Now observe this other image below. We have an A inserted there in a part of set B.

[00:44:17]What does that mean? An intersection.

[00:44:20]So the first image represents a union and the bottom part represents an intersection, okay? We'll see this in the mattresses soon, folks. I'm still waiting for the list to send to you, okay? It's at 97%.

[00:44:37]Let's understand, then, how this issue of union works.

[00:44:41]Okay, I have two sets here, A and B. So, let's say the little boys dressed in red are set A, and the little boys dressed in yellow are set B, okay? So, we have two sets, A and B, and when they are combined, we have the union of sets A and B, right?

[00:45:10]How are we going to represent this in the sets? So, read on, there's that little purple part down there. We're going to have A, this U that indicates the union B, the union with B, okay?

[00:45:26]Perfect. And the intersection? Intersection occurs when we have mixed sets. So, it would be these two here, these three that are here in the middle.

[00:45:36]The red mixed with the yellow. So, it ended up half red, half yellow.

[00:45:42]This is an intersection, okay? So, the intersection will take both sets, set A and set B, but they won't belong to the same set, right? But they will be partly one and partly the other. And then it will be represented as if it were an upside-down U. Look over there in the green part of the slide, A intersects with B, okay? So set A had an intersection with set B. Therefore, it's like a U pointing downwards, union pointing upwards, intersection pointing downwards.

[00:46:26]Right?

[00:46:28]Now let's look at an exercise about this.

[00:46:32]So here I have data and intervals from 3 to 5. Look at the brackets.

[00:46:42]The brackets are facing inwards.

[00:46:44]So that indicates it's a painted ball, right? Full ball. And B, B is the interval that corresponds to the range between 3 and 7 on the number line. However, the number three has the clasp facing outwards. This indicates that the ball is empty. So, he wants you to determine the union of these two sets. So, remember that in a merger, I'm going to take all the items from one group and combine them with all the items from the other group.

[00:47:14]A corresponds to the interval from 3 to 5.

[00:47:19]B, since it has an empty circle, will also correspond to the interval from 3 to 7. However, the number three does not belong to set B, only the number after it. So it will be 4, 5, 6, and 7. In set A, we have 3, 4, and 5. All three belong to set A. So, if I want to combine the two, I'll take three through seven and fill in the circles, right? The little circles need to be filled in. So, A combined with B, then we'll have 3 through 7. So, it corresponds to 3, 4, 5, 6, and 7. All these numbers will belong to this set.

[00:48:08]Quiet?

[00:48:12]And now the intersection.

[00:48:17]So let's look at intersection. What does that mean? Remember what I said here? We're only going to take what corresponds to what we have in common with one and the other, okay? So let's go. Set A, it goes from five, from two to five. However, the brackets are facing outwards.

[00:48:40]So, we're going to have here, look, we're going to put two to five in a straight line, empty circles with empty circles.

[00:48:49]Set B corresponds to 3 A9. The number three is a filled circle because the bracket is facing inwards, and the number nine is a bracket facing outwards. So the ball is empty. Determine the intersection of A and B.

[00:49:08]Well, I've represented it here on the line. Guys, it's important that you always put the numbers in order, on the line, okay? So, look, I have number 2 here, right? 3 4 5.

[00:49:24]In line B, I will have 3 4 5 6 7 8 and 9. Right?

[00:49:32]So, what am I going to do now? To find the intersection, which is this last line, this last straight line, I'm only going to take the numbers that correspond, which I'll have in both A and B, okay? Do I have two or three in B? No.

[00:49:53]So it won't be part of the intersection of three and four. I have it, right?

[00:50:04]And there are also numbers three, four, and five. We have that too, right?

[00:50:13]By the way, five is an open circle, okay?

[00:50:16]So, we're going to keep the number three here with the filled-in circle and the number five with the empty circle.

[00:50:23]We're not going to include anything beyond five because it doesn't correspond; it's not in the A, okay? So it has to be in A and it has to be in B. The only interval that we have in A, that we have in A and that we have in [clearing throat] B, it corresponds from three to five. But five will have an empty circle, right? So, what numbers do we have in this set? The numbers three and four, which are part of this intersection.

[00:50:56]Okay, everyone?

[00:50:59]Let me see how things are going here, folks. It's fully charged, it's going to open, okay? Just wait a little bit.

[00:51:19]Okay everyone, we're going to do this exercise now.

[00:51:26]Represent the intervals on the real number line.

[00:51:31]So you're going to draw the straight line, okay? I'm going to do it with you here on the board.

[00:51:38]Just a second. Let me see if the link opens so I can send it to you. That.

[00:53:30]Just a minute, okay everyone? I'll change the screen for you in a moment. I'm putting this here so we can do an exercise together.

[00:53:48]Oh my gosh, I don't know. I don't think I'll be able to send you that link. My WhatsApp isn't opening and I can't copy the link.

[00:53:58]Let me see if I can do it another way. It did n't freeze, okay guys? I stay quiet. Do you think it crashed?

[00:54:59]Wait a minute. I managed to do it here. I found it easier to send by email. I should have done this sooner.

[00:55:14]Okay, now it's going to work.

[00:55:21]Okay, I sent it. Okay, everyone, I'm just going to switch now to sharing the other screen of the whiteboard so we can do the exercise.

[00:55:38]Oh, can you see the screen yet?

[00:55:48]You can see that, right? So, I've given you the exercises, right? We're going to do these exercises together.

[00:55:54]So, what is this exercise asking us to represent in the form of a straight line?

[00:56:01]So, I'm going to do this first one here, the letter A. What does it say? We have here, look, a bracket sticking out, which indicates to me that the circle is empty, - 2 and 4 with the bracket closed. What does that indicate to me? This indicates to me that the number four belongs to this set. So, the ball is full. So, I'm going to do this representation here for you, okay?

[00:56:28]So, we're going to draw a straight line.

[00:56:30]Oh, that line is so thin. Let me turn it up.

[00:56:37]No, I'll do it with this one.

[00:56:40]Oh, that's too thin. Wait a minute.

[00:56:43]How do I increase this?

[00:56:56]Very thin. No, you have to increase it because there's something here that needs to be erased.

[00:57:22]Ah, I found it here.

[00:57:26]So let's go. It was no use.

[00:57:34]Here is color.

[00:57:38]Well, why did they have to increase the "here"?

[00:57:42]Ah, now it's done.

[00:57:45]So, let's go. Now it will work out.

[00:57:48]So, we're going to represent the letter A, okay, everyone? Letter A. I'll draw the straight line, right? It belongs to the real numbers, right? It says here, look, X belongs to the real numbers. So, I'm going to put here, see, -2 and 4, right? It's not that? The range he wants is from -2 to 4. So, if I have the bracket outside the box, the circle is empty.

[00:58:15]If I have the bracket facing inwards, the little ball gets colored.

[00:58:21]So, look, I represented it there in a straight line, on the real number line. So, here's the letter A, it's done.

[00:58:33]Let's do part B.

[00:58:35]Part B states: X belongs to the real numbers, 1/2 X is less.

[00:58:44]Ah, notice here, x is less than and 2 1/2 less than x less equals 5. So I'm going to represent that on a number line as well.

[00:58:54]So I'll put 1.5 here.

[00:58:57]1.5 people is about the right amount, okay?

[00:59:03]1.5. It means half, it's half, which is the same as 0.5 1/2, and here I have 5, right? When I have that less-than sign, remember? The ball is empty. If I have less than or equal to, the ball is full.

[00:59:30]So, remember, bracket facing outwards, less than sign, empty circle; bracket facing inwards, less than or equal sign, filled circle. Okay, I've already taken that break.

[00:59:46]And this one here, the letter C.

[00:59:52]Wait, let me bring this one a little closer because it's cutting off there.

[01:00:04]And the letter C? The letter C corresponds to infinity and -2. Is this infinity positive or negative?

[01:00:17]Everything here is based on real numbers, okay? It corresponds to the negative. Look, it's less than infinite. So here I have -2 minus infinity, I'm going to put it here on the number line. No, does the little ball have a little ball at infinity? There are no little balls at infinity.

[01:00:39]And we're going to have to put a little ball here. Is this ball full or is this ball empty?

[01:00:47]The ball is empty. Look here. Shovel, the bracket facing outwards. So, I'm going to put this line like this.

[01:00:59]Right?

[01:01:05]So, we've already done the exercise, and we managed to do the intervals, okay, everyone? Were you able to access the link I sent? Click on the link. If you didn't receive it in the group, you'll need to access it through the link I sent in the chat, okay? So you can record your attendance, because from this lesson onwards you will have to keep a record of your attendance, whether it's the online class or the in-person class, okay?

[01:01:42]Hey guys, we missed a few exercises.

[01:01:45]I'll do the following, let me share the other screen here.

[01:01:50]You're going to take a screenshot and do this exercise at home, okay?

[01:01:58]So this is... No, wait a minute.

[01:02:05]Well, he copied it.

[01:02:07]Wait a minute, it's duplicated.

[01:02:11]Exercise two has been duplicated. How crazy.

[01:02:17]Hello.

[01:02:22]Oh, I can't believe he modified it. It really did change.

[01:02:30]Oh, let me check something here, everyone. Wait a minute, I'm going to share the other one with you. I ca n't believe this happened. I think there was some problem with the conversion, because that wasn't my second exercise. Close.

[01:03:07]Hold on, I'm going to open PowerPoint here. Later I'll even proofread it for the in-person class because it came out wrong.

[01:03:19]I don't know if the problem occurred when converting to PDF.

[01:03:23]Sometimes it changes, but I'll confirm here in the PowerPoint presentation I made, because that wasn't the exercise. Oh, worse than that, he looks exactly the same. I must have done something wrong.

[01:03:45]Okay, let's do this, guys? I'll give you the other exercises and you guys will do the others.

[01:03:55]Unfortunately, something went wrong, but I'll fix it later for our class.

[01:04:00]Okay, take a screenshot of this for yourselves, or take a photo of the screen, whichever you think is best, okay?

[01:04:17]I can change this exercise.

[01:04:23]And this exercise, numbers three and four.

[01:04:25]So, numbers three and four will be staying home. And then we'll correct it in the next class, okay? Hey, folks from Thursday, those of you who are here on Thursday, if you want you can try making it at home, right? And then bring it done on Thursday to our in-person class, okay?

[01:05:03]And then I'll give you the other exercise that was missing. So, in our next class, before we start the material, we'll review and correct these exercises, okay?

[01:05:19]Combined? So, guys, it's already been about 5 minutes past our scheduled time, right?

[01:05:26]So, everyone, see you in our next class. Have a good night, have a good week. May God be with you.

[01:05:33]Goodbye. Goodbye.