If in the problem the lengths of the two sides of the triangle and the angle between them are given, then you can apply the formula for the area of a triangle in terms of the sine.
An example of calculating the area of a triangle through a sine. Given the sides a = 3, b = 4, and the angle γ = 30 °. The sine of an angle of 30 ° is 0.5 
The area of the triangle will be 3 square meters. cm.
There may be other conditions as well. If the length of one side and the angles are given, then first you need to calculate the missing angle. Because the sum of all the angles of the triangle is 180 °, then:
The area will be equal to half the square of the side multiplied by the fraction. Its numerator contains the product of the sine of the adjacent angles, and the denominator is the sine of the opposite angle. Now we calculate the area using the following formulas:
For example, given a triangle with side a = 3 and angles γ = 60 °, β = 60 °. We calculate the third angle: 
Substituting data into the formula
We get that the area of the triangle is 3.87 square meters. cm.
II. Area of a triangle in terms of cosine
To find the area of a triangle, you need to know the lengths of all sides. By the cosine theorem, unknown sides can be found, and only then used.
By the cosine theorem, the square of the unknown side of a triangle is equal to the sum of the squares of the remaining sides minus the double product of these sides by the cosine of the angle between them.
From the theorem we derive formulas for finding the length of the unknown side:
Knowing how to find the missing side, having two sides and the angle between them, you can easily calculate the area. The formula for the area of a triangle in terms of the cosine helps you quickly and easily find a solution to various problems.
An example of calculating the formula for the area of a triangle in terms of the cosine
You are given a triangle with known sides a = 3, b = 4, and an angle γ = 45 °. First, find the missing side with... In cosine 45 ° = 0.7. To do this, we substitute the data into the equation derived from the cosine theorem.
Now, using the formula, we find
Area theorem for a triangle
Theorem 1
The area of a triangle is equal to half the product of two sides by the sine of the angle between these sides.
Proof.
Let us be given an arbitrary triangle $ ABC $. Let's denote the lengths of the sides of this triangle as $ BC = a $, $ AC = b $. We introduce a Cartesian coordinate system so that the point $ C = (0,0) $, the point $ B $ lies on the right semiaxis $ Ox $, and the point $ A $ lies in the first coordinate quarter. Let's draw the height $ h $ from the point $ A $ (Fig. 1).
Figure 1. Illustration of Theorem 1
The height of $ h $ is equal to the ordinate of the point $ A $, therefore
Sine theorem
Theorem 2
The sides of a triangle are proportional to the sines of the opposite angles.
Proof.
Let us be given an arbitrary triangle $ ABC $. Let us denote the lengths of the sides of this triangle as $ BC = a $, $ AC = b, $ $ AC = c $ (Fig. 2).
Figure 2.
Let us prove that
By Theorem 1, we have
Equating them in pairs, and we get that
Cosine theorem
Theorem 3
The square of the side of a triangle is equal to the sum of the squares of the other two sides of the triangle without twice the product of these sides by the cosine of the angle between these sides.
Proof.
Let us be given an arbitrary triangle $ ABC $. Let's denote the lengths of its sides as $ BC = a $, $ AC = b, $ $ AB = c $. We introduce a Cartesian coordinate system so that the point $ A = (0,0) $, the point $ B $ lies on the positive semiaxis $ Ox $, and the point $ C $ lies in the first coordinate quarter (Fig. 3).
Figure 3.
Let us prove that
In this coordinate system, we get that
Let us find the length of the side $ BC $ by the formula for the distance between the points
An example of a problem using these theorems
Example 1
Prove that the diameter of the circumscribed circle of an arbitrary triangle is equal to the ratio of any side of the triangle to the sine of the corner opposite to this side.
Solution.
Let us be given an arbitrary triangle $ ABC $. $ R $ - the radius of the circumscribed circle. Let's draw the diameter $ BD $ (Fig. 4).
The area of a triangle is equal to half the product of its sides by the sine of the angle between them.
Proof:
Consider an arbitrary triangle ABC. Let side BC = a, side CA = b and S be the area of this triangle. It is necessary to prove that S = (1/2) * a * b * sin (C).
To begin with, we introduce a rectangular coordinate system and place the origin at point C. Place our coordinate system so that point B lies on the positive direction of the Cx axis, and point A would have a positive ordinate.
If everything is done correctly, you should get the following figure.
The area of a given triangle can be calculated using the following formula: S = (1/2) * a * h where h is the height of the triangle. In our case, the height of the triangle h is equal to the ordinate of point A, that is, h = b * sin (C).
Given the results obtained, the formula for the area of a triangle can be rewritten as follows: S = (1/2) * a * b * sin (C). Q.E.D.
Solving problems
Problem 1. Find the area of a triangle ABC if a) AB = 6 * √8 cm, AC = 4 cm, angle A = 60 degrees b) BC = 3 cm, AB = 18 * √2 cm, angle B = 45 degrees in ) AC = 14 cm, CB = 7 cm, angle C = 48 degrees.
By the theorem proved above, the area S of a triangle ABC is equal to:
S = (1/2) * AB * AC * sin (A).
Let's make the calculations:
a) S = ((1/2) * 6 * √8 * 4 * sin (60˚)) = 12 * √6 cm ^ 2.
b) S = (1/2) * BC * BA * sin (B) = ((1/2) * 3 * 18 * √2 * (√2 / 2)) = 27 cm ^ 2.
c) S = (1/2) * CA * CB * sin (C) = ½ * 14 * 7 * sin48˚ cm ^ 2.
The value of the sine of the angle is calculated on a calculator or we use the values from the table of values trigonometric angles... Answer:
a) 12 * √6 cm ^ 2.
c) approximately 36.41 cm ^ 2.
Problem 2. The area of triangle ABC is 60 cm ^ 2. Find side AB if AC = 15 cm, angle A = 30˚.
Let S be the area of triangle ABC. By the theorem on the area of a triangle, we have:
S = (1/2) * AB * AC * sin (A).
Let's substitute our existing values into it:
60 = (1/2) * AB * 15 * sin30˚ = (1/2) * 15 * (1/2) * AB = (15/4) * AB.
From here we express the length of the side AB: AB = (60 * 4) / 15 = 16.
Simply put, these are vegetables cooked in water according to a special recipe. I will consider two initial components (vegetable salad and water) and the finished result - borscht. Geometrically, this can be thought of as a rectangle with one side representing lettuce and the other side representing water. The sum of these two sides will represent borscht. The diagonal and area of such a "borscht" rectangle are purely mathematical concepts and are never used in borscht recipes.
How do lettuce and water turn into borscht from a mathematical standpoint? How can the sum of two line segments turn into trigonometry? To understand this, we need linear angle functions.
You won't find anything about linear angle functions in mathematics textbooks. But without them there can be no mathematics. The laws of mathematics, like the laws of nature, work regardless of whether we know about their existence or not.
Linear angle functions are addition laws. See how algebra turns into geometry and geometry turns into trigonometry.
Can linear angle functions be dispensed with? You can, because mathematicians still do without them. The trick of mathematicians lies in the fact that they always tell us only about those problems that they themselves know how to solve, and never talk about those problems that they cannot solve. Look. If we know the result of addition and one term, we use subtraction to find the other term. Everything. We do not know other tasks and are not able to solve them. What to do if we only know the result of addition and do not know both terms? In this case, the result of the addition must be decomposed into two terms using linear angle functions. Then we ourselves choose what one term can be, and the linear angle functions show what the second term should be so that the result of the addition is exactly what we need. There can be an infinite number of such pairs of terms. In everyday life, we perfectly manage without the decomposition of the sum, subtraction is enough for us. But in scientific research of the laws of nature, the decomposition of the sum into terms can be very useful.
Another law of addition, which mathematicians don't like to talk about (another trick of theirs), requires that the terms have the same units of measurement. For salad, water and borscht, these can be units of measure for weight, volume, value, or units of measure.
The figure shows two levels of difference for math. The first level is the differences in the field of numbers, which are indicated a, b, c... This is what mathematicians do. The second level is the differences in the area of units, which are shown in square brackets and indicated by the letter U... This is what physicists do. We can understand the third level - differences in the area of the described objects. Different objects can have the same number of identical units of measure. How important this is, we can see on the example of borscht trigonometry. If we add subscripts to the same designation of units of measurement of different objects, we can say exactly which mathematical value describes a particular object and how it changes over time or in connection with our actions. By letter W I will designate water, with the letter S I will designate the salad and the letter B- Borsch. This is what the linear angular functions for borsch would look like.
If we take some part of the water and some part of the salad, together they will turn into one portion of borscht. Here I suggest you take a break from borscht and remember your distant childhood. Remember how we were taught to put bunnies and ducks together? It was necessary to find how many animals there would be. What then were we taught to do? We were taught to separate units from numbers and add numbers. Yes, any number can be added to any other number. This is a direct path to the autism of modern mathematics - we are doing it is not clear what, it is not clear why, and we very poorly understand how this relates to reality, because of the three levels of difference, mathematics operates only one. It would be more correct to learn how to switch from one measurement unit to another.
And bunnies, and ducks, and animals can be counted in pieces. One common unit of measurement for different objects allows us to add them together. This is a childish version of the problem. Let's take a look at a similar problem for adults. What happens when you add bunnies and money? There are two possible solutions here.
First option... We determine the market value of the bunnies and add it to the available amount of money. We got the total value of our wealth in monetary terms.
Second option... You can add the number of bunnies to the number of banknotes we have. We will receive the number of movable property in pieces.
As you can see, the same addition law produces different results. It all depends on what exactly we want to know.
But back to our borscht. Now we can see what will happen for different values of the angle of the linear angle functions.
The angle is zero. We have salad, but no water. We cannot cook borscht. The amount of borscht is also zero. This does not mean at all that zero borscht is equal to zero water. Zero borscht can be at zero salad (right angle).
For me personally, this is the main mathematical proof of the fact that. Zero does not change the number when added. This is because the addition itself is impossible if there is only one term and there is no second term. You can relate to this as you like, but remember - all mathematical operations with zero were invented by mathematicians themselves, so discard your logic and stupidly cram definitions invented by mathematicians: "division by zero is impossible", "any number multiplied by zero equals zero" , "for the knock-out point zero" and other nonsense. It is enough to remember once that zero is not a number, and you will never have a question whether zero is a natural number or not, because such a question generally loses any meaning: how can we consider a number that is not a number. It's like asking what color an invisible color should be. Adding zero to a number is like painting with paint that doesn't exist. We waved with a dry brush and told everyone that "we have painted". But I digress a little.
Injection Above zero, but less than forty-five degrees. We have a lot of salad, but not enough water. As a result, we get a thick borscht.
The angle is forty-five degrees. We have equal amounts of water and salad. This is the perfect borscht (yes, the cooks will forgive me, it's just math).
The angle is greater than forty-five degrees, but less than ninety degrees. We have a lot of water and little salad. You get liquid borscht.
Right angle. We have water. From the salad, only memories remain, as we continue to measure the angle from the line that once stood for the salad. We cannot cook borscht. The amount of borscht is zero. In that case, hold on and drink the water while you have it)))
Here. Something like this. I can tell other stories here that will be more than appropriate here.
Two friends had their shares in the common business. After killing one of them, everything went to the other.
The emergence of mathematics on our planet.
All of these stories are told in the language of mathematics using linear angle functions. Some other time I'll show you the real place of these functions in the structure of mathematics. In the meantime, let's return to the trigonometry of the borscht and consider the projections.
Saturday, 26 October 2019
I watched an interesting video about Grandi row One minus one plus one minus one - Numberphile... Mathematicians lie. They did not perform the equality test in the course of their reasoning.
This echoes my reasoning about.
Let's take a closer look at the signs of deceiving us by mathematicians. At the very beginning of reasoning, mathematicians say that the sum of the sequence DEPENDS on whether the number of elements in it is even or not. This is an OBJECTIVELY DETERMINED FACT. What happens next?
Then mathematicians subtract a sequence from one. What does this lead to? This leads to a change in the number of elements in the sequence - an even number changes to an odd number, an odd number changes to an even number. After all, we have added one element equal to one to the sequence. Despite all the external similarities, the sequence before conversion is not equal to the sequence after conversion. Even if we are talking about an infinite sequence, we must remember that an infinite sequence with an odd number of elements is not equal to an infinite sequence with an even number of elements.
Putting an equal sign between two sequences differing in the number of elements, mathematicians argue that the sum of the sequence DOES NOT DEPEND on the number of elements in the sequence, which contradicts an OBJECTIVELY DETERMINED FACT. Further reasoning about the sum of an infinite sequence is false, since it is based on false equality.
If you see that mathematicians in the course of proofs place parentheses, rearrange the elements of a mathematical expression, add or remove something, be very careful, most likely they are trying to deceive you. Like card magicians, mathematicians distract your attention with various expression manipulations in order to end up slipping you a false result. If you cannot repeat the card trick without knowing the secret of deception, then in mathematics everything is much simpler: you do not even suspect anything about deception, but repeating all the manipulations with a mathematical expression allows you to convince others of the correctness of the result, just like when something convinced you.
Question from the audience: And what about infinity (as the number of elements in sequence S), is it even or odd? How can you change the parity of something that has no parity?
Infinity for mathematicians, like the Kingdom of Heaven for priests - no one has ever been there, but everyone knows exactly how everything works there))) I agree that after death you will be absolutely indifferent whether you have lived an even or odd number of days, but ... just one day at the beginning of your life, we will get a completely different person: his surname, name and patronymic are exactly the same, only the date of birth is completely different - he was born one day before you.
And now, in essence))) Suppose a finite sequence that has parity loses this parity when going to infinity. Then any finite segment of an infinite sequence must also lose parity. We do not see this. The fact that we cannot say for sure whether the number of elements in an infinite sequence is even or odd does not mean at all that parity has disappeared. Parity, if it exists, cannot disappear without a trace into infinity, as in the sleeve of a sharper. There is a very good analogy for this case.
Have you ever asked a cuckoo sitting in a clock in which direction the clock hand rotates? For her, the arrow rotates in the opposite direction to what we call "clockwise". As paradoxical as it sounds, the direction of rotation depends solely on which side we observe the rotation from. And so, we have one wheel that turns. We cannot say in which direction the rotation occurs, since we can observe it both from one side of the plane of rotation, and from the other. We can only attest to the fact that there is rotation. Complete analogy with the parity of an infinite sequence S.
Now let's add a second spinning wheel, whose plane of rotation is parallel to the plane of rotation of the first spinning wheel. We still cannot say for sure in which direction these wheels rotate, but we can absolutely say for sure whether both wheels rotate in the same direction or in opposite directions. Comparing two endless sequences S and 1-S, I showed with the help of mathematics that these sequences have different parity and putting an equal sign between them is an error. Personally, I believe in mathematics, I do not trust mathematicians))) By the way, for a complete understanding of the geometry of transformations of infinite sequences, it is necessary to introduce the concept "simultaneity"... This will need to be drawn.
Wednesday, 7 August 2019
Concluding the conversation about, there is an infinite number to consider. The result is that the concept of "infinity" acts on mathematicians like a boa constrictor on a rabbit. The trembling horror of infinity deprives mathematicians of common sense. Here's an example:
The original source is located. Alpha stands for real number... The equal sign in the above expressions indicates that if you add a number or infinity to infinity, nothing will change, the result will be the same infinity. Taking as an example the infinite set natural numbers, then the considered examples can be presented as follows:
For a visual proof of their correctness, mathematicians have come up with many different methods. Personally, I look at all these methods as dancing shamans with tambourines. Essentially, they all boil down to the fact that either some of the rooms are not occupied and new guests are moving in, or that some of the visitors are thrown out into the corridor to make room for guests (very humanly). I presented my view on such decisions in the form of a fantastic story about the Blonde. What is my reasoning based on? Relocating an infinite number of visitors takes an infinite amount of time. After we have vacated the first room for a guest, one of the visitors will always walk along the corridor from his room to the next one until the end of the century. Of course, the time factor can be stupidly ignored, but it will already be from the category "the law is not written for fools." It all depends on what we are doing: adjusting reality to match mathematical theories or vice versa.
What is an "endless hotel"? An endless hotel is a hotel that always has any number of vacant places, no matter how many rooms are occupied. If all the rooms in the endless visitor corridor are occupied, there is another endless corridor with the guest rooms. There will be an endless number of such corridors. Moreover, the "infinite hotel" has an infinite number of floors in an infinite number of buildings on an infinite number of planets in an infinite number of universes created by an infinite number of Gods. Mathematicians, however, are not able to distance themselves from commonplace everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. Here are mathematicians and are trying to manipulate the serial numbers of hotel rooms, convincing us that it is possible to "shove the stuff in."
I will demonstrate the logic of my reasoning to you on the example of an infinite set of natural numbers. First, you need to answer a very simple question: how many sets of natural numbers are there - one or many? There is no correct answer to this question, since we invented numbers ourselves, in Nature there are no numbers. Yes, Nature is excellent at counting, but for this she uses other mathematical tools that are not familiar to us. As Nature thinks, I will tell you another time. Since we invented the numbers, we ourselves will decide how many sets of natural numbers there are. Consider both options, as befits a real scientist.
Option one. "Let us be given" a single set of natural numbers, which lies serenely on the shelf. We take this set from the shelf. That's it, there are no other natural numbers left on the shelf and there is nowhere to take them. We cannot add one to this set, since we already have it. And if you really want to? No problem. We can take one from the set we have already taken and return it to the shelf. After that, we can take a unit from the shelf and add it to what we have left. As a result, we again get an infinite set of natural numbers. You can write all our manipulations like this:
I wrote down the actions in the algebraic notation system and in the notation system adopted in set theory, with a detailed enumeration of the elements of the set. The subscript indicates that we have one and only set of natural numbers. It turns out that the set of natural numbers will remain unchanged only if one subtracts from it and adds the same unit.
Option two. We have many different infinite sets of natural numbers on our shelf. I emphasize - DIFFERENT, despite the fact that they are practically indistinguishable. We take one of these sets. Then we take one from another set of natural numbers and add it to the set we have already taken. We can even add two sets of natural numbers. Here's what we get:
Subscripts "one" and "two" indicate that these items belonged to different sets. Yes, if you add one to the infinite set, the result will also be an infinite set, but it will not be the same as the original set. If we add another infinite set to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.
Lots of natural numbers are used for counting in the same way as a ruler for measurements. Now imagine adding one centimeter to the ruler. This will already be a different line, not equal to the original.
You can accept or not accept my reasoning - it's your own business. But if you ever run into mathematical problems, think about whether you are not following the path of false reasoning trodden by generations of mathematicians. After all, doing mathematics, first of all, form a stable stereotype of thinking in us, and only then add mental abilities to us (or, on the contrary, deprive us of free thought).
pozg.ru
Sunday, 4 August 2019
I was writing a postscript to an article about and saw this wonderful text on Wikipedia:
We read: "... the rich theoretical foundation of Babylonian mathematics did not have a holistic character and was reduced to a set of disparate techniques, devoid of a common system and evidence base."
Wow! How smart we are and how well we can see the shortcomings of others. Is it hard for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, I personally got the following:
The rich theoretical basis of modern mathematics is not holistic and is reduced to a set of disparate sections devoid of a common system and evidence base.
I will not go far to confirm my words - it has a language and conventions that are different from the language and conventions of many other branches of mathematics. The same names in different branches of mathematics can have different meanings. I want to devote a whole series of publications to the most obvious blunders of modern mathematics. See you soon.
Saturday, 3 August 2019
How do you divide a set into subsets? To do this, it is necessary to enter a new unit of measurement that is present for some of the elements of the selected set. Let's look at an example.
Let us have many A consisting of four people. This set was formed on the basis of "people" Let us denote the elements of this set by the letter a, a subscript with a digit will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sex" and denote it by the letter b... Since sexual characteristics are inherent in all people, we multiply each element of the set A by gender b... Note that now our multitude of "people" has become a multitude of "people with sex characteristics." After that, we can divide the sex characteristics into masculine bm and women bw sexual characteristics. Now we can apply a mathematical filter: we select one of these sex characteristics, it does not matter which one is male or female. If a person has it, then we multiply it by one, if there is no such sign, we multiply it by zero. And then we apply the usual school mathematics. See what happened.
After multiplication, reduction and rearrangement, we got two subsets: the subset of men Bm and a subset of women Bw... Mathematicians think about the same when they apply set theory in practice. But they do not devote us to the details, but give a finished result - "a lot of people consist of a subset of men and a subset of women." Naturally, you may wonder how correctly the mathematics is applied in the above transformations? I dare to assure you, in fact, the transformations were done correctly, it is enough to know the mathematical basis of arithmetic, Boolean algebra and other branches of mathematics. What it is? Some other time I'll tell you about it.
As for supersets, you can combine two sets into one superset by choosing the unit of measurement that is present for the elements of these two sets.
As you can see, units of measurement and common mathematics make set theory a thing of the past. An indication that set theory is not all right is that mathematicians have come up with their own language and notation for set theory. Mathematicians did what shamans once did. Only shamans know how to "correctly" apply their "knowledge". They teach us this "knowledge".
Finally, I want to show you how mathematicians manipulate
Let's say Achilles runs ten times faster than a turtle and is a thousand steps behind it. During the time it takes Achilles to run this distance, the turtle will crawl a hundred steps in the same direction. When Achilles has run a hundred steps, the turtle will crawl ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the turtle.
This reasoning came as a logical shock to all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them has become a generally accepted solution to the question ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from magnitude to. This transition implies application instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement either has not yet been developed, or it has not been applied to Zeno's aporia. Applying our usual logic leads us into a trap. We, by inertia of thinking, apply constant units of measurement of time to the reciprocal. From a physical point of view, it looks like time dilation until it stops completely at the moment when Achilles is level with the turtle. If time stops, Achilles can no longer overtake the turtle.
If we turn over the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of his path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly catch up with the turtle."
How can you avoid this logical trap? Stay in constant time units and do not go backwards. In Zeno's language, it looks like this:
During the time during which Achilles will run a thousand steps, the turtle will crawl a hundred steps in the same direction. Over the next interval of time, equal to the first, Achilles will run another thousand steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred steps ahead of the turtle.
This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insuperability of the speed of light is very similar to the Zeno aporia "Achilles and the Turtle". We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia Zeno tells about a flying arrow:
The flying arrow is motionless, since at every moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow rests at different points in space, which, in fact, is motion. Another point should be noted here. From a single photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs are needed, taken from the same point at different points in time, but it is impossible to determine the distance from them. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What do I want to turn Special attention, so it is that two points in time and two points in space are different things that should not be confused, because they provide different opportunities for research.
Let me show you the process with an example. We select "red solid in a pimple" - this is our "whole". At the same time, we see that these things are with a bow, but there are no bows. After that we select a part of the "whole" and form a set "with a bow". This is how shamans feed themselves by tying their set theory to reality.
Now let's do a little dirty trick. Take "solid in a pimple with a bow" and combine these "wholes" by color, selecting the red elements. We got a lot of "red". Now a question to fill in: the resulting sets "with a bow" and "red" are the same set or are they two different sets? Only shamans know the answer. More precisely, they themselves do not know anything, but as they say, so be it.
This simple example shows that set theory is completely useless when it comes to reality. What's the secret? We have formed a set of "red solid into a bump with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a pimple), ornaments (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics... This is what it looks like.
The letter "a" with different indices denotes different units measurements. Units of measurement are highlighted in brackets, by which the "whole" is allocated at the preliminary stage. The unit of measurement, by which the set is formed, is taken out of the brackets. The last line shows the final result - the element of the set. As you can see, if we use units of measurement to form a set, then the result does not depend on the order of our actions. And this is mathematics, not the dancing of shamans with tambourines. Shamans can "intuitively" arrive at the same result, arguing it "by the obviousness", because units of measurement are not included in their "scientific" arsenal.
It is very easy to use units to split one or combine several sets into one superset. Let's take a closer look at the algebra of this process.
