Hotkeys occupy an important place among the ways to speed up the interaction with the computer. Thanks to them, we get access to the desired function almost instantly, instead of wandering through the menu items for a long time and hitting them with the mouse. Therefore, hotkeys are equally useful for both beginners and experienced users. On the pages of MacRadar, we have repeatedly raised the topic of hotkeys. In this article, I will talk about modifier keys that cover various areas of application and how to directly enter popular special characters.

Note. As for the input of special characters, some of them need to be entered in the English layout, since in Russian there will be completely different characters.

Mathematical symbols

For pupils, students, researchers and in general all those who often have to fiddle with equations and mathematical symbols on their Macs, it will be very useful to know how to enter them directly from the keyboard without resorting to a bank of symbols or replacing them with similar ones (like m3 or<1). Ввод символов напрямую с клавиатуры довольно удобная вещь, которая здорово экономит время.

1. Inequality sign ≠

To insert a math symbol ≠ click ⌥ = .

2. Plus-minus sign ±

To enter a character ± - click ⇧⌥ = (English layout) or ⇧ ⌥§ (Russian).

3. Infinity sign ∞

If you need to put the symbol ∞ - click ⌥ 5 (English layout).

4. Ellipsis...

You don't need three dots to insert an ellipsis - just press ⌥ ; (English layout).

5. Division sign ÷

To get this symbol ÷ - press ⌥ / (English layout).

6. Greater than or equal sign ≥

To insert a greater than or equal to symbol, press ⌥ > .

7. Less than or equal sign ≤

To get the opposite symbol ≤ - press ⌥ < .

8. Pi sign π

The number π is often found in equations and races, if you need to enter it - click ⌥ P(English layout).

Working with screenshots

9. Screenshot of the whole screen

To take a screenshot of the entire screen, click ⌘ ⇧ 3 . The screenshot will automatically be saved to your desktop.

10. Screenshot of the screen area

In this case, click ⌘ ⇧ 4 and without releasing the keys, select the desired area of ​​the screen.

11. Screenshot of a specific window

Sometimes you need to take a screenshot of a separate window, for this click ⌘ ⇧ 4 then Spacebar and click. (after pressing the spacebar, you can move between windows to select the one you need).

12. Copy screenshot to clipboard

All screenshots are automatically saved to the desktop, but if you are anxious about the order on it and do not allow clutter - just add the key to the above combinations ⌃ . That is, ⌘ ⇧ 4⌃ takes a screenshot of the selected window and copies it to the clipboard.

Entering special characters

Using the keyboard, you can enter not only the characters printed on the keys, but many other useful characters associated with a particular key. Here are some popular symbols that you might find useful.

13. Trademark™

If you need to enter the icon ™ trade mark - click ⌥ 2 .

14.Registered Trademark®

To enter a registered trademark - click ⌥ R.

15. Copyright ©

Click ⌥ G to get the copyright symbol.

16. Euro currency symbol €

To enter the euro symbol, press ⌥⇧ 2 .

17. Bulleted List Item

You can quickly create a neat bulleted list by clicking ⌥ 8 on each line.

18. Paragraph symbol ¶

If you need to specify a paragraph symbol, press ⌥ 7.

19. Dagger (footnote symbol) †

Click ⌥ T to insert a character denoting a footnote.

20. Degree º

Click ⌥ 0 to enter a degree.

21. Greek letters delta, beta and omega ∂ ß Ω

If you need to enter the letters of the Greek alphabet ∂ , ß , Ω - click ⌥ D, ⌥ S, ⌥ Z, respectively.

System boot, shutdown

While booting a Mac, you can use different keys for a particular type of boot. Here is some of them.

22. Show boot disks

Holding ⌥ during boot, you can display all available boot disks.

23. Boot in safe mode

Hold down the key to boot into safe mode ⇧ .

24. Booting from an external drive

Sometimes it is necessary to boot from an external source: USB, DVD - to do this, hold down the key FROM.

25. Recovery mode (recovery)

To boot into recovery mode, hold the combination ⌘ R.

26. Download in Single User Mode

Click ⌘ S in order to boot into this mode.

27. Sleep mode

When you press ⌘⌥⏏ your Mac will go to sleep.

28. Calling the shutdown/reboot menu

pressing ⌃ ⏏ will open the standard shutdown/reboot/sleep dialog.

Hotkeys for Shopping Cart

Deleting files can be done in different ways, but the easiest way to do this is with shortcuts. There are also combinations for emptying and completely emptying the Recycle Bin. About them further.

29. Deleting files

To delete the selected files, click ⌘⌫ . On large keyboards where there is a key ⌦ , you can press ⌘⌦ .

30. File recovery

To restore selected files from the Recycle Bin, you need to press the same combination ⌘⌫ (⌘⌦ ).

31. Emptying the Recycle Bin

To empty the Recycle Bin, click ⌘ ⇧ ⌫ in Finder. After that, you need to confirm the deletion.

32. Emptying the Trash (no confirmation)

To empty the Recycle Bin without prompting you to confirm deletion, click ⌘⌥ ⇧ ⌫ (⌘⌥ ⇧ ⌦ ).

33. Bonus

To insert the Apple logo  use the shortcut ⌥ ⇧ K.

If you liked working with hotkeys, I recommend that you familiarize yourself with the previous collections that were published on MacRadar.

• 50+ Useful Safari Productivity Keyboard Shortcuts

As always, your comments are welcome, dear readers. Tell us about your favorite shortcuts - we're always happy to hear your opinion!

Hotkeys occupy an important place among the ways to speed up the interaction with the computer. Thanks to them, we get access to the desired function almost instantly, instead of wandering through the menu items for a long time and hitting them with the mouse. Therefore, hotkeys are equally useful for both beginners and experienced users. On the pages of MacRadar, we have repeatedly raised the topic of hotkeys. In this article, I will talk about modifier keys that cover various areas of application and how to directly enter popular special characters.

Note. As for the input of special characters, some of them need to be entered in the English layout, since in Russian there will be completely different characters.

Mathematical symbols

For pupils, students, researchers and in general all those who often have to fiddle with equations and mathematical symbols on their Macs, it will be very useful to know how to enter them directly from the keyboard without resorting to a bank of symbols or replacing them with similar ones (like m3 or<1). Ввод символов напрямую с клавиатуры довольно удобная вещь, которая здорово экономит время.

1. Inequality sign ≠

To insert a math symbol ≠ click ⌥ = .

2. Plus-minus sign ±

To enter a character ± - click ⇧⌥ = (English layout) or ⇧ ⌥§ (Russian).

3. Infinity sign ∞

If you need to put the symbol ∞ - click ⌥ 5 (English layout).

4. Ellipsis...

You don't need three dots to insert an ellipsis - just press ⌥ ; (English layout).

5. Division sign ÷

To get this symbol ÷ - press ⌥ / (English layout).

6. Greater than or equal sign ≥

To insert a greater than or equal to symbol, press ⌥ > .

7. Less than or equal sign ≤

To get the opposite symbol ≤ - press ⌥ < .

8. Pi sign π

The number π is often found in equations and races, if you need to enter it - click ⌥ P(English layout).

Working with screenshots

9. Screenshot of the whole screen

To take a screenshot of the entire screen, click ⌘ ⇧ 3 . The screenshot will automatically be saved to your desktop.

10. Screenshot of the screen area

In this case, click ⌘ ⇧ 4 and without releasing the keys, select the desired area of ​​the screen.

11. Screenshot of a specific window

Sometimes you need to take a screenshot of a separate window, for this click ⌘ ⇧ 4 then Spacebar and click. (after pressing the spacebar, you can move between windows to select the one you need).

12. Copy screenshot to clipboard

All screenshots are automatically saved to the desktop, but if you are anxious about the order on it and do not allow clutter - just add the key to the above combinations ⌃ . That is, ⌘ ⇧ 4⌃ takes a screenshot of the selected window and copies it to the clipboard.

Entering special characters

Using the keyboard, you can enter not only the characters printed on the keys, but many other useful characters associated with a particular key. Here are some popular symbols that you might find useful.

13. Trademark™

If you need to enter the icon ™ trade mark - click ⌥ 2 .

14.Registered Trademark®

To enter a registered trademark - click ⌥ R.

15. Copyright ©

Click ⌥ G to get the copyright symbol.

16. Euro currency symbol €

To enter the euro symbol, press ⌥⇧ 2 .

17. Bulleted List Item

You can quickly create a neat bulleted list by clicking ⌥ 8 on each line.

18. Paragraph symbol ¶

If you need to specify a paragraph symbol, press ⌥ 7.

19. Dagger (footnote symbol) †

Click ⌥ T to insert a character denoting a footnote.

20. Degree º

Click ⌥ 0 to enter a degree.

21. Greek letters delta, beta and omega ∂ ß Ω

If you need to enter the letters of the Greek alphabet ∂ , ß , Ω - click ⌥ D, ⌥ S, ⌥ Z, respectively.

System boot, shutdown

While booting a Mac, you can use different keys for a particular type of boot. Here is some of them.

22. Show boot disks

Holding ⌥ during boot, you can display all available boot disks.

23. Boot in safe mode

Hold down the key to boot into safe mode ⇧ .

24. Booting from an external drive

Sometimes it is necessary to boot from an external source: USB, DVD - to do this, hold down the key FROM.

25. Recovery mode (recovery)

To boot into recovery mode, hold the combination ⌘ R.

26. Download in Single User Mode

Click ⌘ S in order to boot into this mode.

27. Sleep mode

When you press ⌘⌥⏏ your Mac will go to sleep.

28. Calling the shutdown/reboot menu

pressing ⌃ ⏏ will open the standard shutdown/reboot/sleep dialog.

Hotkeys for Shopping Cart

Deleting files can be done in different ways, but the easiest way to do this is with shortcuts. There are also combinations for emptying and completely emptying the Recycle Bin. About them further.

29. Deleting files

To delete the selected files, click ⌘⌫ . On large keyboards where there is a key ⌦ , you can press ⌘⌦ .

30. File recovery

To restore selected files from the Recycle Bin, you need to press the same combination ⌘⌫ (⌘⌦ ).

31. Emptying the Recycle Bin

To empty the Recycle Bin, click ⌘ ⇧ ⌫ in Finder. After that, you need to confirm the deletion.

32. Emptying the Trash (no confirmation)

To empty the Recycle Bin without prompting you to confirm deletion, click ⌘⌥ ⇧ ⌫ (⌘⌥ ⇧ ⌦ ).

33. Bonus

To insert the Apple logo  use the shortcut ⌥ ⇧ K.

If you liked working with hotkeys, I recommend that you familiarize yourself with the previous collections that were published on MacRadar.

• 50+ Useful Safari Productivity Keyboard Shortcuts

As always, your comments are welcome, dear readers. Tell us about your favorite shortcuts - we're always happy to hear your opinion!

Alpha denotes a 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. If we take an infinite set of natural numbers as an example, then the considered examples can be represented as follows:

To visually prove their case, mathematicians have come up with many different methods. Personally, I look at all these methods as the dances of shamans with tambourines. In essence, they all come down to the fact that either some of the rooms are not occupied and new guests are settled in them, or that some of the visitors are thrown out into the corridor to make room for the 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? Moving an infinite number of visitors takes an infinite amount of time. After we have vacated the first guest room, one of the visitors will always walk along the corridor from his room to the next one until the end of time. Of course, the time factor can be stupidly ignored, but this will already be from the category of "the law is not written for fools." It all depends on what we are doing: adjusting reality to mathematical theories or vice versa.

What is an "infinite hotel"? An infinity inn is an inn that always has any number of vacancies, no matter how many rooms are occupied. If all the rooms in the endless hallway "for visitors" are occupied, there is another endless hallway with rooms for "guests". There will be an infinite number of such corridors. At the same time, 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, on the other hand, are not able to move away from banal everyday problems: God-Allah-Buddha is always only one, the hotel is one, the corridor is only one. So mathematicians are trying to juggle the serial numbers of hotel rooms, convincing us that it is possible to "shove the unpushed".

I will demonstrate the logic of my reasoning to you using the example of an infinite set of natural numbers. First you need to answer a very simple question: how many sets of natural numbers exist - one or many? There is no correct answer to this question, since we ourselves invented numbers, there are no numbers in Nature. Yes, Nature knows how to count perfectly, 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 exist. Consider both options, as befits a real scientist.

Option one. "Let us be given" a single set of natural numbers, which lies serenely on a 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. What if you really want to? No problem. We can take a unit 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 recorded the actions in algebraic system notation and in the system of notation 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 is subtracted from it and the same one is added.

Option two. We have many different infinite sets of natural numbers on the 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:

The subscripts "one" and "two" indicate that these elements belonged to different sets. Yes, if you add one to an infinite set, the result will also be an infinite set, but it will not be the same as the original set. If another infinite set is added to one infinite set, the result is a new infinite set consisting of the elements of the first two sets.

The set of natural numbers is used for counting in the same way as a ruler for measurements. Now imagine that you have added 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 - this is your own business. But if you ever run into mathematical problems, consider whether you are on the path of false reasoning, trodden by generations of mathematicians. After all, mathematics classes, first of all, form a stable stereotype of thinking in us, and only then they add to us mental capacity(or vice versa, deprive us of free thought).

Sunday, August 4, 2019

I was writing a postscript to an article about and saw this wonderful text on Wikipedia:

We read: "... rich theoretical background mathematics of Babylon did not have a holistic character and was reduced to a set of disparate techniques, devoid of common system and evidence base.

Wow! How smart we are and how well we can see the shortcomings of others. Is it weak for us to look at modern mathematics in the same context? Slightly paraphrasing the above text, personally I got the following:

The rich theoretical basis of modern mathematics does not have a holistic character 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 cycle of publications to the most obvious blunders of modern mathematics. See you soon.

Saturday, August 3, 2019

How to divide a set into subsets? To do this, you must enter a new unit of measure, which is present in some of the elements of the selected set. Consider an example.

May we have many BUT consisting of four people. This set is formed on the basis of "people" Let's designate the elements of this set through the letter a, the subscript with a number will indicate the ordinal number of each person in this set. Let's introduce a new unit of measurement "sexual characteristic" and denote it by the letter b. Since sexual characteristics are inherent in all people, we multiply each element of the set BUT on gender b. Notice that our "people" set has now become the "people with gender" set. After that, we can divide the sexual characteristics into male bm and women's bw gender characteristics. Now we can apply a mathematical filter: we select one of these sexual characteristics, it does not matter which one is male or female. If it is present in a person, 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, reductions and rearrangements, we got two subsets: the male subset bm and a subset of women bw. Approximately the same way mathematicians reason when they apply set theory in practice. But they do not let us in on the details, but give us the finished result - "a lot of people consists of a subset of men and a subset of women." Naturally, you may have a question, how correctly applied mathematics in the above transformations? I dare to assure you that in fact the transformations are done correctly, it is enough to know the mathematical justification of arithmetic, Boolean algebra and other sections of mathematics. What it is? Some other time I will tell you about it.

As for supersets, it is possible to combine two sets into one superset by choosing a unit of measurement that is present in the elements of these two sets.

As you can see, units of measurement and common math make set theory a thing of the past. A sign that all is not well with set theory is that mathematicians have come up with their own language and notation for set theory. The mathematicians did what the shamans once did. Only shamans know how to "correctly" apply their "knowledge". This "knowledge" they teach us.

Finally, I want to show you how mathematicians manipulate .

Monday, January 7, 2019

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

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... 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 became a universally accepted solution to the problem ..."[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 the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. 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 time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

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 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 overtake the tortoise."

How to avoid this logical trap? stay in constant units measurements of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one 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 to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet 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, since at each 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 movement. There is another point to be noted here. From one 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 taken from the same point at different points in time are needed, but they cannot be used to determine the distance. 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 (naturally, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

I already told you that, with the help of which shamans try to sort "" realities. How do they do it? How does the formation of the set actually take place?

Let's take a closer look at the definition of a set: "a collection of different elements, conceived as a single whole." Now feel the difference between the two phrases: "thinkable as a whole" and "thinkable as a whole." The first phrase is the end result, the multitude. The second phrase is a preliminary preparation for the formation of the set. At this stage, reality is divided into separate elements ("whole") from which a multitude ("single whole") will then be formed. At the same time, the factor that allows you to combine the "whole" into a "single whole" is carefully monitored, otherwise the shamans will not succeed. After all, shamans know in advance exactly what set they want to demonstrate to us.

I will show 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, and there are without a bow. 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 trick. Let's take "solid in a pimple with a bow" and unite these "whole" by color, selecting red elements. We got a lot of "red". Now a tricky question: are the received sets "with a bow" and "red" the same set or 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 formed a set of "red solid pimply with a bow". The formation took place according to four different units of measurement: color (red), strength (solid), roughness (in a bump), decorations (with a bow). Only a set of units of measurement makes it possible to adequately describe real objects in the language of mathematics. Here's what it looks like.

The letter "a" with different indices denotes different units of measurement. In parentheses, units of measurement are highlighted, according to which the "whole" is allocated at the preliminary stage. The unit of measurement, according to which the set is formed, is taken out of brackets. The last line shows the final result - an element of the set. As you can see, if we use units to form a set, then the result does not depend on the order of our actions. And this is mathematics, and not the dances of shamans with tambourines. Shamans can “intuitively” come to the same result, arguing it with “obviousness”, because units of measurement are not included in their “scientific” arsenal.

With the help of units of measurement, it is very easy to break one or combine several sets into one superset. Let's take a closer look at the algebra of this process.

Saturday, June 30, 2018

If mathematicians cannot reduce a concept to other concepts, then they do not understand anything in mathematics. I answer: how do the elements of one set differ from the elements of another set? The answer is very simple: numbers and units of measurement.

It is today that everything that we do not take belongs to some set (as mathematicians assure us). By the way, did you see in the mirror on your forehead a list of those sets to which you belong? And I have not seen such a list. I will say more - not a single thing in reality has a tag with a list of sets to which this thing belongs. Sets are all inventions of shamans. How do they do it? Let's look a little deeper into history and see how the elements of the set looked before the mathematicians-shamans pulled them apart into their sets.

A long time ago, when no one had heard of mathematics yet, and only trees and Saturn had rings, huge herds of wild elements of sets roamed the physical fields (after all, shamans had not yet invented mathematical fields). They looked like this.

Yes, do not be surprised, from the point of view of mathematics, all elements of sets are most similar to sea ​​urchins- from one point, like needles, units of measurements stick out in all directions. For those who, I remind you that any unit of measurement can be geometrically represented as a segment of arbitrary length, and a number as a point. Geometrically, any quantity can be represented as a bundle of segments sticking out in different sides from one point. This point is the zero point. I will not draw this work of geometric art (no inspiration), but you can easily imagine it.

What units of measurement form an element of the set? Any that describe this element from different points of view. These are the ancient units of measurement used by our ancestors and which everyone has long forgotten about. These are the modern units of measurement that we use now. These are units of measurement unknown to us, which our descendants will come up with and which they will use to describe reality.

We figured out the geometry - the proposed model of the elements of the set has a clear geometric representation. And what about physics? Units of measurement - this is the direct connection between mathematics and physics. If shamans do not recognize units of measurement as a full-fledged element of mathematical theories, this is their problem. I personally can’t imagine a real science of mathematics without units of measurement. That is why, at the very beginning of the story about set theory, I spoke of it as the Stone Age.

But let's move on to the most interesting - to the algebra of elements of sets. Algebraically, any element of the set is a product (the result of multiplication) of different quantities. It looks like this.

I deliberately did not use the conventions adopted in set theory, since we are considering an element of a set in a natural habitat before the advent of set theory. Each pair of letters in brackets denotes a separate value, consisting of the number indicated by the letter " n" and units of measurement, indicated by the letter " a". Indexes near the letters indicate that the numbers and units of measurement are different. One element of the set can consist of an infinite number of values ​​\u200b\u200b(as long as we and our descendants have enough imagination). Each bracket is geometrically represented by a separate segment. In the example with the sea urchin one bracket is one needle.

How do shamans form sets from different elements? In fact, by units of measurement or by numbers. Understanding nothing in mathematics, they take different sea urchins and carefully examine them in search of that single needle by which they form a set. If there is such a needle, then this element belongs to the set; if there is no such needle, this element is not from this set. Shamans tell us fables about mental processes and a single whole.

As you may have guessed, the same element can belong to a variety of sets. Next, I will show you how sets, subsets and other shamanistic nonsense are formed. As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

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

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the entire amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. 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 for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it 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, without any "conceivable as not a single whole" or "not conceivable as a single whole."

HTML special characters are special language constructs that refer to characters from the character set used in text files. The table below lists the reserved and special characters that cannot be added to the source code of an HTML document using the keyboard:

• characters that cannot be entered using the keyboard (for example, the copyright symbol)
• symbols intended for markup (for example, a greater than or less than sign)

Such characters are added using a numeric code or a name.

Symbol	Numeric code	Symbol name	Description
"	"	"	quotation mark
"	"	"	apostrophe
&	&	&	ampersand
<		<	less sign
>	>	>	greater sign
			non-breaking space (A non-breaking space is a space that appears inside a line as a regular space, but prevents display and printing programs from breaking the line at that point.)
¡	¡	¡	inverted exclamation mark
¢	¢	¢	cent
£	£	£	lb.
¤	¤	¤	currencies
¥	¥	¥	yen
¦	¦	¦	broken vertical bar
§	§	§	section
¨	¨	¨	interval (cyrillic)
	©		copyright sign
ª	ª	ª	female ordinal index
«	«	«	French quotation marks (Christmas trees) - left
¬	¬	¬	negation-expressions
®	®	®	registered trademark
¯	¯	¯	macron interval
°	°	°	degree
±	±	±	plus or minus
²	²	²	superscript 2
³	³	³	superscript 3
´	´	´	acute interval
µ	µ	µ	micro
¶	¶	¶	paragraph
·	·	·	middle point
¸	¸	¸	interval cedilla
¹	¹	¹	superscript 1
º	º	º	male ordinal index
»	»	»	French quotation marks (Christmas trees) - right
¼	¼	¼	1/4 part
½	½	½	1/2 part
¾	¾	¾	3/4 parts
¿	¿	¿	upside down question mark
×	×	×	multiplication
÷	÷	÷	division
́	́		stress
Œ	Œ	Œ	ligature uppercase OE
œ	œ	œ	lowercase ligature oe
Š	Š	Š	S with crown
š	š	š	lowercase S with crown
Ÿ	Ÿ	Ÿ	capital Y with tiara
ƒ	ƒ	ƒ	f with hook
ˆ	ˆ	ˆ	dicriatic accent
˜	˜	˜	little tilde
–	–	-	dash
—	—	—	em dash
‘	‘	‘	left single quote
’	’	’	right single quote
‚	‚	‚	bottom single quote
“	“	“	left double quotes
”	”	”	right double quotes
„	„	„	bottom double quotes
†	†	†	dagger
‡	‡	‡	double dagger
		.	bullet
…	…	…	horizontal ellipsis
‰	‰	‰	ppm (thousandths)
′	′	′	minutes
″	″	″	seconds
‹	‹	‹	single left angle quote
›	›	›	single right angle quote
‾	‾	‾	overlining
€	€	€	Euro
™	™ or	™	trademark
←	←	←	left arrow
			up arrow
→	→	→	right arrow
↓	↓	↓	arrow to down
↔	↔	↔	double sided arrow
↵	↵	↵	carriage return arrow
⌈	⌈	⌈	upper left corner
⌉	⌉	⌉	top right corner
⌊	⌊	⌊	lower left corner
⌋	⌋	⌋	bottom right corner
◊	◊	◊	rhombus
♠	♠	♠	peaks
♣	♣	♣	baptize
			worms
♦	♦	♦	bubi

Mathematical Symbols Supported in HTML

Symbol	Numeric code	Symbol name	Description
∀	∀	∀	for anyone, for everyone
∂	∂	∂	part
∃	∃	∃	exists
∅	∅	∅	empty set
∇	∇	∇	Hamilton operator ("nabla")
∈	∈	∈	belongs to the set
∉	∉	∉	does not belong to the set
∋	∋	∋	or
∏	∏	∏	work
∑	∑	∑	sum
−	−	−	minus
∗	∗	∗	multiplication or operator adjoint to
×	×	×	multiplication sign
√	√	√	Square root
∝	∝	∝	proportionality
∞	∞	∞	infinity
⋮	⋮		multiplicity
∠	∠	∠	corner
∧	∧	∧	and
∨	∨	∨	or
∩	∩	∩	intersection
∪	∪	∪	an association
∫	∫	∫	integral
∴	∴	∴	that's why
∼	∼	∼	like
≅	≅	≅	comparable
≈	≈	≈	approximately equal to
≠	≠	≠	not equal
≡	≡	≡	identically
≤	≤	≤	less or equal
⩽	⩽ ⩽	⩽ ⩽	less or equal
≥	≥	≥	more or equal
⩾	⩾ ⩾	⩾ ⩾	more or equal
⊂	⊂	⊂	subset
⊃	⊃	⊃	supersets
⊄	⊄	⊄	not a subset
⊆	⊆	⊆	subset
⊇	⊇	⊇	superset
⊕	⊕	⊕	direct sum
⊗	⊗	⊗	tenzer product
⊥	⊥	⊥	perpendicular
⋅	⋅	⋅	dot operator

Greek and Coptic alphabets

Symbol	Numeric code	Hex code	Symbol name
Ͱ	Ͱ	Ͱ
ͱ	ͱ	ͱ
Ͳ	Ͳ	Ͳ
ͳ	ͳ	ͳ
ʹ	ʹ	ʹ
͵	͵	͵
Ͷ	Ͷ	Ͷ
ͷ	ͷ	ͷ
ͺ	ͺ	ͺ
ͻ	ͻ	ͻ
ͼ	ͼ	ͼ
ͽ	ͽ	ͽ
;	;	;
΄	΄	΄
΅	΅	΅
Ά	Ά	Ά
·	·	·
Έ	Έ	Έ
Ή	Ή	Ή
Ί	Ί	Ί
Ό	Ό	Ό
Ύ	Ύ	Ύ
Ώ	Ώ	Ώ
ΐ	ΐ	ΐ
Α	Α	Α	Α
Β	Β	Β	Β
Γ	Γ	Γ	Γ
Δ	Δ	Δ	Δ
Ε	Ε	Ε	Ε
Ζ	Ζ	Ζ	Ζ
Η	Η	Η	Η
Θ	Θ	Θ	Θ
Ι	Ι	Ι	Ι
Κ	Κ	Κ	Κ
Λ	Λ	Λ	Λ
Μ	Μ	Μ	Μ
Ν	Ν	Ν	Ν
Ξ	Ξ	Ξ	Ξ
Ο	Ο	Ο	Ο
Π	Π	Π	Π
Ρ	Ρ	Ρ	Ρ
Σ	Σ	Σ	Σ
Τ	Τ	Τ	Τ
Υ	Υ	Υ	Υ
Φ	Φ	Φ	Φ
Χ	Χ	Χ	Χ
Ψ	Ψ	Ψ	Ψ
Ω	Ω	Ω	Ω
Ϊ	Ϊ	Ϊ
Ϋ	Ϋ	Ϋ
ά	ά	ά
έ	έ	έ
ή	ή	ή
ί	ί	ί
ΰ	ΰ	ΰ
α	α	α	α
β	β	β	β
γ	γ	γ	γ
δ	δ	δ	δ
ε	ε	ε	ε
ζ	ζ	ζ	ζ
η	η	η	η
θ	θ	θ	θ
ι	ι	ι	ι
κ	κ	κ	κ
λ	λ	λ	λ
μ	μ	μ	μ
ν	ν	ν	ν
ξ	ξ	ξ	ξ
ο	ο	ο	ο
π	π	π	π
ρ	ρ	ρ	ρ
ς	ς	ς	ς
σ	σ	σ	σ
τ	τ	τ	τ
υ	υ	υ	υ
φ	φ	φ	φ
χ	χ	χ	χ
ψ	ψ	ψ	ψ
ω	ω	ω	ω
ϊ	ϊ	ϊ
ϋ	ϋ	ϋ
ό	ό	ό
ύ	ύ	ύ
ώ	ώ	ώ
Ϗ	Ϗ	Ϗ
ϐ	ϐ	ϐ
ϑ	ϑ	ϑ	ϑ
ϒ	ϒ	ϒ	ϒ
ϓ	ϓ	ϓ
ϔ	ϔ	ϔ
ϕ	ϕ	ϕ	ϕ
ϖ	ϖ	ϖ	ϖ
ϗ	ϗ	ϗ
Ϙ	Ϙ	Ϙ
ϙ	ϙ	ϙ
Ϛ	Ϛ	Ϛ
ϛ	ϛ	ϛ
Ϝ	Ϝ	Ϝ	Ϝ
ϝ	ϝ	ϝ	ϝ
Ϟ	Ϟ	Ϟ
ϟ	ϟ	ϟ
Ϡ	Ϡ	Ϡ
ϡ	ϡ	ϡ
Ϣ	Ϣ	Ϣ
ϣ	ϣ	ϣ
Ϥ	Ϥ	Ϥ
ϥ	ϥ	ϥ
Ϧ	Ϧ	Ϧ
ϧ	ϧ	ϧ
Ϩ	Ϩ	Ϩ
ϩ	ϩ	ϩ
Ϫ	Ϫ	Ϫ
ϫ	ϫ	ϫ
Ϭ	Ϭ	Ϭ
ϭ	ϭ	ϭ
Ϯ	Ϯ	Ϯ
ϯ	ϯ	ϯ
ϰ	ϰ	ϰ	ϰ
ϱ	ϱ	ϱ	ϱ
ϲ	ϲ	ϲ
ϳ	ϳ	ϳ
ϴ	ϴ	ϴ
ϵ	ϵ	ϵ	ϵ
϶	϶	϶	϶
Ϸ	Ϸ	Ϸ
ϸ	ϸ	ϸ
Ϲ	Ϲ	Ϲ
Ϻ	Ϻ	Ϻ
ϻ	ϻ	ϻ
ϼ	ϼ	ϼ
Ͻ	Ͻ	Ͻ
Ͼ	Ͼ	Ͼ
Ͽ	Ͽ	Ͽ

Why special characters are needed and how to use them

Suppose you decide to describe some tag on your page, but since the browser uses characters< и >like a start and end tag, applying them inside your html content can lead to problems. But HTML gives you an easy way to define these and other special characters with simple abbreviations called symbol references.

Let's see how it works. For each character that is considered special or that you want to use on your web page but that cannot be printed in your editor (for example, a copyright character), you find an abbreviation and print it in the html code instead of the desired character. For example, for the symbol ">", the abbreviation is - > , and for the symbol "<" - < .

Let's say you wanted to print "Element very important" on its page. Instead, you will have to use references to the symbols you need to correctly display the entry, and as a result, your entry in the code should look like this:

Element very important

Try »

Another special character you need to be aware of is the & (ampersand) symbol. If you want it to appear on your HTML page, use the & reference instead of the & character.

Along with arithmetic operations, there is an acquaintance with such abstract concepts as "greater than", "less than" and "equal to". It will not be difficult for a child to determine which side has more objects and which one has less. But here the setting of signs sometimes causes difficulties. Game methods will help to learn the signs.

"Hungry Bird"

To play, you will need a sign - an open beak (a "more" sign). It can be cut out of cardboard or made into a large model from a disposable plate. To interest the baby, you can glue or draw eyes, feathers, and make the mouth open .

The explanation starts with some background: “This bird is small, loves to eat well. And she always chooses the pile in which there is more food.

After that, it is clearly shown that the bird opens its beak to the side where there are more objects.

Further, the information received is fixed: heaps with grains are laid out on the table, and the child determines in which direction the bird will turn its beak . If it is not possible to correctly position it the first time, you need to help by saying again that the mouth is open towards more food. Then you can offer several more similar tasks: the numbers are written on the sheet, you need to glue the beak correctly.

Examples can be diversified by replacing the bird with a pike, a crocodile, or any other predator that also opens its mouth towards a larger number.

There may be unusual situations where the number of items in both piles will be equal. If the child notices this, it means that he is attentive.

For this you must be commended , and then show 2 identical strips and explain that they are the same as the number of objects in piles, and since the number of objects is equal, then the sign is called “equal”.

Arrows

A small schoolchild can be explained signs based on comparing them with arrows pointing in different directions.

Difficulties may arise when reading expressions. But this difficulty can also be overcome: by putting the sign correctly, he will be able to correctly read the expression . After completing a few exercises, the child will remember that the arrow pointing to the left means the sign "less". If she points to the right, then the sign reads: "more."

Strengthening exercises

After explaining the rules for setting the sign, you need to practice in performing similar tasks.

For this purpose, tasks of this type are suitable:

1. "Put a sign" (4 and 5 - need a "less than" sign).
2. "More less" - the child shows signs with the thumb and forefinger of both hands, comparing the sizes of various objects or their number (the plane is larger than the dragonfly, the strawberry is smaller than the watermelon).
3. "What number" - there are signs, a number is written on one side, you need to guess what number will be on the other side (in the expression "_<5» на месте пропуска могут стоять числа 0 – 4).
4. "Fill in the numbers" - you need to correctly put the numbers to the left and right of the specified sign (the number 8 will be to the left of the "greater than" sign, and the number 2 to the right).

To develop logic and thinking, you can supplement the exercises with the following tasks:

• "From which direction did the object escape?" - 3 triangles are drawn on the left, 2 squares on the right, and there is a “=” sign between them. The child must guess that there is not enough square on the right for the equality to be true. If you can’t do this right away, you can solve the problem practically by adding a triangle first on the left, and then a square on the right.
• “What needs to be done to make inequality right?” - taking into account the situation, the child determines which side to remove or add objects so that the sign stands correctly.

Video tutorial will tell you about the signs: greater than, less than and equal