Set and multiset

The concepts of “set” and “multiset” are two trump ace in the sleeves of a sharpie, one in each sleeve. The sharpie takes out from the sleeve that trump ace, which is more convenient for him to get. In any case, the sharpie always wins. In the same way, any mathematician will always justify set theory.

There can’t be two identical elements in a set, but if there are identical elements in a set, such a set is called a “multiset.” Intelligent beings can never understand such a logic of absurdity. This is the level of talking parrots and trained monkeys. Mathematicians act as ordinary animal trainer, preaching to us their absurd ideas.

Once the engineers who built the bridge, during the testing of the bridge, were in a boat under the bridge. If the bridge collapsed, a mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, a talented engineer built other bridges.

No matter how mathematicians hide behind the phrase “mathematics studies abstract concepts”, there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. We apply the mathematical set theory to the mathematicians themselves.

We studied mathematics very well and now we are giving out salaries. Here comes a mathematician for his money. We count the whole amount to him and lay out on his table on different piles, into which we place banknotes of the same denomination. Then we take one banknote from each pile and hand over to the mathematician his "mathematical set of salary." We explain the mathematics that he will receive the remaining bills only when he proves that a set without the same elements is not equal to a set with the same elements. After that, the most interesting will happen.

Set and multiset. Money. Mathematics for blondes.
Set and multiset

First of all, the logic of the deputies will work: "It can be applied to others, but not to me!" Further, the mathematician will assure us that on banknotes of the same denomination there are different numbers of banknotes, therefore they cannot be considered the same elements of the set. Well, we count the salary in coins - there are no numbers on the coins. Here, the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms of each coin is unique ...

And now I have the most interesting question: where is the line behind which the elements of the multiset turn into the elements of the set and vice versa? Such a facet does not exist - shamans decide everything, science is absent here.

You see that the same set of elements is both a multitude and a multiset at the same time. How right? And here the mathematician-shaman-schuller takes an ace from his sleeve and begins to tell us either about the multitude or the multiset. In any case, he will convince us that the theory of sets is correct.

To understand how modern shamans apply set theory to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you how to do it.

Set theory

Set theory is the stone age of mathematics. Only shamans know how to properly form the set. Let's look at set theory from the side and reveal some of the secrets of shamans. To understand the essence of set theory, it is necessary to consider another mathematical concept - a function. The function sets the correspondence between the elements of two sets. Each element from the herd of hunters is associated with one or more elements from the herd of prey, each element from the herd of prey is assigned one or more elements from the herd of hunters. Only shamans know what kind of herd belongs to and how to divide prey properly.

What is set theory? This is a herd theory. What will happen if we combine a herd (set) mathematicians and a herd (set) sheep: sheeps with a mathematical education or mathematicians with sheep’s brains? I don’t know what set theory says about the results of such a union, but in reality mathematicians will have a great barbecue picnic. I have nothing against set theory, as one of the mathematical tools. But to stand up for “scientific” theories based on such a primitive and vague concept is already too much.

Any theory should be tested by practice, even mathematical. I will show you an example of the practical application of set theory by the example of such concepts as “set” and “multiset”.


Achilles and the tortoise

Achilles and the tortoise. Mathematics For Blondes.
Achilles and the tortoise

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporia, the most famous of which is the Achilles and the Tortoise aporia. Here’s how it sounds:

Suppose Achilles runs ten times faster than a turtle, and is a thousand paces behind it. During the time that Achilles has run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred paces, the tortoise crawls ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

Another version of the aporia "Achilles and the tortoise":

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This reasoning was a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them in one way or another viewed the aporia of Zeno. The shock turned out to be so strong that discussions are continuing at the present time; the scientific community has not yet succeeded in reaching a general opinion on the essence of paradoxes. Mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the question; none of them has become a generally accepted solution. Everyone understands that this is a hoax, but no one understands what it is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from quantity to an inverse quantity. This transition implies the use of variable units of measurement instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed or has not been applied to the aporia of Zeno. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like a slowing down of time to its full stop at the moment when Achilles approaches the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming is ten times less than the previous one. If you use the concept of "infinity" in this situation, then it will be correct to say "Achilles will catch up to the tortoise infinitely quickly".

How to avoid this logical trap? Stay in constant time units and don’t go to inverse quantity. In the language of Zeno, it looks like this:

In the time that Achilles has run a thousand steps, the tortoise will crawl a hundred steps in the same direction. Over the next interval of time equal to the first, Achilles will run a thousand more steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution. On the aporia of Zeno "Achilles and the Turtle" is very similar to Einstein's statement about the irresistible speed of light. 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 of Zeno tells of a flying arrow:

A flying arrow is motionless, because at each moment of time it is motionless, and since it is motionless at each moment of time, it is always motionless.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment in time a flying arrow is stationary at different points in space, which, in fact, is movement.



In Wikipedia there is a page of a trapezoid. In the drawing it is shown that any trapezoid can be turned into a rectangle.

Trapezoid. Mathematics For Blondes.
Let's look at algebra. These are formulas for calculation of lengths of diagonals of a trapezoid.

Diagonals of trapezoid. Mathematics For Blondes.
Diagonals of trapezoid
We substitute in these formulas data for a rectangle.

Diagonals of rectangle. Mathematics For Blondes.
Diagonals of rectangle
If to trust these formulas, the rectangle has no diagonals. Even schoolboys or schoolgirls can make what mathematicians couldn't make – to execute verification of the solution of a task. It is the actual level of modern mathematics – any statement of mathematicians can be false.

Height of a trapezoid

Height of a trapezoid is determined by the area of a triangle. The area of a triangle is calculated on Heron's formula. The sizes of the parties of a trapezoid allow to receive a triangle which has the same height as a trapezjid. The cunning trick of mathematicians allows to calculate length of diagonals of a trapezoid.

Height of a trapezoid. Mathematics For Blondes.
Height of a trapezoid
 When the legs of a trapezoid are parallel, the triangle disappears and the cunning trick ceases to work. If to determine height of a trapezoid by the area of a trapezoid, then no problems arise upon transition to a rectangle or a parallelogram.

Height of a rectangle. Mathematics For Blondes.
Height of a rectangle
 Conclusion: cunning tricks of mathematicians can result in false results.


Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

Inverse transformations. We transformed the sum of a line segments to the law of cosines. Mathematics For Blondes.
Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

Measurement of a angle. 0 degree and 180 degrees. Mathematics For Blondes.
Measurement of a angle
It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

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We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

The Pythagorean theorem and the sum of a line segments. Mathematics For Blondes.
The Pythagorean theorem and the sum of a line segments
In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

Right triangle and two a line segments. Mathematics For Blondes.
Right triangle and two a line segments
How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

Triangle and the law of cosines. Mathematics For Blondes.
Triangle and the law of cosines
Right triangle and the Pythagorean theorem. Law of cosines. Mathematics For Blondes.
Right triangle and the Pythagorean theorem
The law of cosines and sum of two a line segments. Mathematics For Blondes.
The law of cosines and sum of two a line segments
We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

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Law of sines

The law of sines (sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

The law of sines. Sine law, sine formula, or sine rule. Mathematics For Blondes.
The law of sines
a, b, c - are the lengths of the sides of a triangle;
α, β, γ - are the opposite angles;
T - are the area of triangle;
R - are the radius of the triangle's circumcircle.

How to use this monster? Use dress-making courses. Cut the law of sines on a part.

Cut the law of sines on a part. Mathematics For Blondes.
Cut the law of sines on a part
Make a necessary formula of two parts. Use properties of proportions.

Example of use of the law of sine. Mathematics For Blondes.
Example of use of the law of sine