Kіntseva podmnozhina. The meaning of the word pіdmnozhina. Por_vnyalny analysis of the capabilities of people and machines

On a simple example, we guess what is called a submultiple, how to use a submultiplier (high and low), a formula for the significance of the number of all submultiples, as well as a calculator, which looks like an impersonal multiplier.

example 1. An impersonal A = (a, c, p, o) is given. Write down the numbers
given multiples.
Solution:

Vlasni pіdnіzhini:(a), (c), (p), (o), (a, c), (a, p), (a, o), (c, p), (c, o) ∈, (p, o), (a, c, p), (a, c, o), (c, p, o).

Naughty:(a, s, p, pro), Ø.

Total: 16 submultiples.

Explanation. The multiplier A is submultiplied by the multiplier B, so the skin element of the multiplier A is also offset by B.

An empty multiplier ∅ є submultiple, be it a multiplicity, is called unsettled;
. be-like a multiplicity є submultiple to itself, so it is called inconspicuous;
. Whether an n-element multiplier has exactly 2 n submultiples.

Remaining firm formula for knowing the number of all submultiples without pererahuvannya skin.

Visnovok formula: Suppose we have є impersonal s n-elements. When folding the first element, the first element can lie down, or lie down, tobto. the first element can be selected in two ways, similarly for all other elements (all n-elements), the skin can be selected in two ways, and following the multiplication rule it is possible: 2∙2∙2∙ ...∙2=2 n

For mathematicians, we formulate a theorem and induce a suvori proof.

Theorem. The number of submultiples of the final multiplier, which is added up from n elements, is 2 n.

Bringing. An impersonal thing that is made up of one element a, can be two (tobto 2 1) multiplication: ∅ that (a). An impersonal, which is composed of two elements a and b, maє chotiri (tobto 2 2) multiply: ∅, (a), (b), (a; b).
The multiplier that is added up from three elements a, b, c, maє vіsіm (tobto 2 3) submultiple:
∅, (a), (b), (b; a), (c), (c; a), (c; b), (c; b; a).
You can let it go, that adding a new element to the stock of a new number of submultiples.
Let us complete the proof of the proof by the method of mathematical induction. The essence of this method is that it is true for a certain cob natural number n 0 and that it is true for a sufficient natural number n \u003d k ≥ n 0, it is possible to bring it true for the number k + 1, then the goal of power is fair for all natural numbers.

1. For n = 1 (base of induction) (and then n = 2, 3) the theorem is completed.

2. Assume that the theorem is confirmed for n = k, then. the number of submultipliers that add up from the elements, add up to 2 k.

3. Let's say that the number of submultipliers of B, which is added up with n = k + 1 elements, adds up to 2 k+1.
We choose the real element b of the multiplier B. Let's look at the impersonal A = B \ (b). It will take revenge on the elements k. All submultipliers of A - all submultipliers of B, so as not to avenge element b i, for omissions, їх 2 k pieces. Multiply the multiplier B to replace the element b, the columns w, tobto. 2 k
things.

Then, all submultiplies of B: 2 k + 2 k = 2 ⋅ 2 k = 2 k + 1 pieces.
The theorem has been completed.

At the butt 1 faceless A \u003d (a, c, p, o) it is added up with four elements, n=4, then, the number of all additional elements is 24 =16.

If you need to write down all the submultiples, or create a program for writing impersonal all submultiples, then there is an algorithm for solving: it is possible to represent combinations of two numbers. Let's explain in an example.

butt 2.Є impersonal (ab c)
000 = (0) (empty multiple)
001=(c)
010 = (b)
011 = (bc)
100 = (a)
101 = (a c)
110 = (a b)
111 = (a b c)

Anonymous calculator

The calculator already has the elements of the multiplier A \u003d (a, c, p, o), just press the Submit button. If you need the perfection of your task, then we collect the elements of the multiplier in Latin, through whom, as shown in the butt.

Nalezhni $A$, also lie down $B$. Formal appointment:

$(A\; \backslash subset\; B)\; \backslash Leftrightarrow\; \backslash forall\; x.\; (x\; \backslash in\; A\; \backslash Rightarrow\; x\; \backslash in\; B).$

Bezlich $B$ called over the multiplier faceless $A$, like $A$- submultiple $B$.

Use two symbolic designations for submultiples:

Offended by the system, the meaning of the vicorist symbol $\backslash subset$ in other senses, which can lead to a swindler. In these articles, we will win the rest of the designation system.

Those who $B$ is called a multiplicity $A$, often write $B\; \backslash supset\; A$.

The impersonality of all submultiples $A$ be appointed $\backslash mathcal(P)(A)$ and is called boolean.

Vlasna pіdnіzhina

Be-yaka impersonal $B$є his pіdnіzhinoyu. Yakscho mi wanto turn off $B$ at a glance wet

Bezlich $A$є vlasniy podmnozhinoy mnozhini $B$, like $A\backslash subset\; B$і $A\; \backslash ne\; B$.

An empty multiplier is a submultiple, be it a multiplier. Likewise, we also want to turn off the sight of the empty faceless, we respect the understanding non-trivial Multiply what is displayed like this:

Bezlich $A$є non-trivial multiplier $B$, like $A$є vlasniy pіdnіzhinoy $B$і $A\; \backslash ne\backslash varnothing$.

Apply

• Bezlich $\backslash varnothing,\; \backslash (0\backslash ),\; \backslash (1,3,4\backslash ).$ $\backslash \{\; 0,1,2,3,4,5\backslash \}$
• Bezlich $\backslash (\backslash varnothing,\; \backslash uparrow,\; moose\backslash ),\; \backslash (\$,\%,*,\backslash uparrow\backslash ),\; \backslash (\backslash varnothing\backslash ),\; \backslash varnothing$є submultiple multiplier $\backslash (\$,\; \%,\; \backslash varnothing,\; \backslash uparrow,\; *,\; moose\; \backslash )$
• Come on $A\; =\; \backslash (a,b\backslash )$ also $\backslash mathcal(P)(A)\; =\; \backslash (\backslash varnothing,\; \backslash (a\backslash ),\; \backslash (b\backslash ),\; \backslash (a,b\backslash )\; \backslash )$.
• Come on $A\; =\; \backslash \; (1,2,3,4,5\; \backslash ),\; \backslash ;\; B\; =\; \backslash \; (1,2,3\; \backslash ),\; \backslash ;\; c=\backslash (4,5,6,7)$. Todi $B\; \backslash subset\; A,\backslash ;\; C\backslash not\backslash subset\; A$.

power

The establishment of a multiplicity of maє tsіlu low authorities.

• Vіdnoshennia podmnozhini є stavlennyam chastkovy order:
  • Suggestions are multiplied reflexively: $B\backslash subset\; B$
  • Variation of multiplier is antisymmetric: $(A\; \backslash subset\; B\; \backslash ;\; \backslash and\; \backslash ;\; B\; \backslash subset\; A)\; \backslash Leftrightarrow\; (A\; =\; B)$
  • The submultiplier setting is transitive: $(A\; \backslash subset\; B\; \backslash ;\backslash and\; \backslash ;\; B\; \backslash subset\; C)\; \backslash Rightarrow\; (A\; \backslash subset\; C)$
• An empty multiplicity is a multiplicity of whether there is something else, to that there is a smaller multiplicity of a multiplicity: $\backslash varnothing\backslash subset\; B$
• For some kind of two multiplies $A$і $B$ offensive hardness equivalent:
  • $A\backslash subsetB.$
  • $A\backslash capB=A.$
  • $A\; \backslash cup\; B\; =\; B.$
  • $B^(\backslash complement)\; \backslash subset\; A^(\backslash complement).$

Multiply kіtsevyh multiplies

If you have a lot of money, then the new one has a lot of money. And yourself, at $n$-element multiplier $2^n$ podmnozhin (including empty). To perekonatisya in tsoma, dosit remember that the skin element can enter or not enter into the submultiplier, but to mean, Zagalna kіlkіst pіdnіzhin bude $n$-multiple creation of two. How can I see less than a dozen $n$-element multiplier $k\backslash le\; n$ elements, their number is expressed by a binomial coefficient $\backslash textstyle\backslash binom(n)(k)$. To reconcile this fact, you can choose elements and add multiples sequentially. The first element can be selected $n$ ways, other $n-1$ way, and so far, i, nareshti, $k$-th element can be selected $n-k+1$ way. In this rank, we take away the sequence of $k$ elements, and equally $k!$ such sequences are supported by one submultiple. It means to find $\backslash textstyle\backslash frac(n(n-1)\backslash dots(n-k+1))(k=\backslash binom(n)(k)!\}$ such submultiples.

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Notes

Literature

• Vereshchagin N. K., Shen A. Lectures on mathematical logic and theory of algorithms. Part 1. Cobs of the theory of multiples. - 3rd species, stereotype. - M.: MTSNMO, 2008. - 128 p. - ISBN 978-5-94057-321-0.

formal Mathematical (theoretical, symbolic) 2 constants: 0 1 Div. also

Lesson that characterizes the submultiple

- It's not my fault that Rozmova went for other officers. Possibly, it is not necessary to speak in front of them, I am not a diplomat. I sweat at the hussars and pishov, thinking that it’s not necessary to be thin here, but it seems to me that I’m lying ... then let me be satisfied ...
- Tse everything is good, no one thinks that you are a boyaguz, that’s not the point. Ask Denisov, it looks like the price is right, so the junker was satisfied with the regimental commander?
Denisov, biting his nose, listening to Rozmov with a frown with a frown, might not hesitate to enter into a new one. To supply the headquarters of the captain of the wines, he snatched his head in a snarky way.
- You tell the regimental commander about the guide in front of the officers, - they gave the headquarters of the captain. - Bogdanovich (Bogdanovich was called the regimental commander) overlaying you.
- Not overlaying, but saying that I'm telling a lie.
- Well, you have said foolish things to you, and you need to ask again.
- No way! shouted Rostov.
- I didn’t think about what kind of you, - the headquarters captain said seriously and suvoro. - You don’t want to get out, but you, father, not only in front of him, but in front of our regiment, in front of us, you’re all around wine. And from yak: yakbi you thought she was glad, how to deal with her on the right, otherwise you straight, she was for the officers, and thumped. What is the job of the regimental commander now? Is it necessary to witness the court of an officer and confuse the entire regiment? For one negidnik the whole regiment will be slandered? So, what do you think? But in our opinion, not so. Well done Bogdanovich, having told you that you are telling a lie. It’s unacceptable, that scho, father, they ran into it themselves. And now, as if you really want to hush it up, so you don’t want to speak out because of the fanaticism, but you want to tell everything. You're covered, you're on duty, the one who vibes in front of the old and honest officer! What a bi there is not buv Bogdanovich, but all honest and good, old colonel, so cover for you; is it okay for you to wander the regiment? - The voice of the headquarters of the captain began three. - Vee, father, the regiment has no fate; ninі here, tomorrow they went to the adjutant; you don’t give a damn what they will say: “Mij by Pavlograd officers of villainy!” And we don't care. So what, Denisov? Not all the same?
Denisov kept mumbling and not breaking in, glancing at Rostov with his gleaming black eyes.
“Your own fancy is dear to you, you don’t want to leave,” continued the captain’s headquarters, “but we, old, as we grew up, die, God willing, join the regiment, then the honor of the regiment is dear to us, and Bogdanovich knows. Oh, how dear, tattoo! And tse not good, not good! There, appear chi ni, and I will tell the truth to the uterus. Not good!
The 1st headquarters of the captain moved and returned to Rostov.
- Pg "avda, chog" take it! - Shouting, hoarse, Denisov. - Well, G "brush! Well!
Rostov, red-eyed, marveling first at one, then at another officer.
- No, sir, no ... don’t think ... I’ll be more reasonable, you’ll think so about me ... I ... for me ... I’m for the honor of the regiment ... that scho? I’ll show you the truth, and for me the honor of an ensign ... well, it’s all the same, really, I’m guilty! .. - Tears stood in Yogo’s eyes. - I'm guilty, all around guilty! ... Well, what else do you need? ...
“Otse so, count,” turning around, muttering the headquarters of the captain, hitting yoga. great hand on the shoulder.
- I'll tell you, - shouting Denisov, - win small glorious.
- So shorter, count, - repeating the headquarters of the captain, nibi for yoga recognition, start calling him with a title. - Go and vibachtesya, your Excellency, that s.
“Panov, I’m ruining everything, I don’t feel a word from me,” after Rostov was promoted in a benevolent voice, “but I can’t shake it off, by God, I can’t, if you want! How do I vibe, like a little one, ask for forgiveness?
Denisov laughed.
- You're worse off. Bogdanovich remembers evil, to pay for being stuck, - saying Kirsten.
- Oh God, don't get stuck! I can’t describe to you, like I can’t almost…
- Well, your will, - said the staff captain. - Well, where did the bastard go? - I asked for wine from Denisov.
- Giving signs of illness, tomorrow ordered pkazin to turn off, - having promoted Denisov.
- Tse ailment, otherwise it is not possible to explain, - having said the headquarters of the captain.
- Even there, the ailment is not the ailment, but don’t get caught in my eyes - I’ll kill you! Denisov shouted bloodily.
To the kimnati zaishov Zherkov.
- You yak? - the officers turned back in a rapt to the one who killed.
- Go away, sir. The poppy is full of building and for the army, zovsim.
- Breshesh!
- I'm a bachiv.
- Yak? Maca live bachiv? with arms, with legs?
- Get out! Let's go! Give you a dance for such a novelty. Have you eaten here?
- Znov was sent to the regiment for the devil, for Mack. The Austrian general shrugged. I have adopted yoga with the arrival of Mack ... Quiet, Rostov, from the lazne?
- Here, brother, we have such porridge another day.
Regimental adjutant and having confirmed the call, I brought it to Zherkov. For tomorrow it was ordered to speak out.
- Get out, sir!
- Well, thank God, we sat up.

Kutuzov stepped up to Sunday, building bridges behind him on the rivers Inn (near Braunau) and Trauni (near Linci). 23 days. Russian warriors crossed the river Ens. Russian convoys, artillery and military columns in the middle of the day dragged across the Ens place, stars and from that side of the bridge.

In rich multitudes, one can see other groups of elements, uniting their own power. For example, for impersonal natural numbers, you can see a submultiple of paired numbers, as well as a submultiple of unpaired numbers, or a submultiple of numbers not more than 100 and so on.

In the terminology of the theory, multiplies seem to be that the multiplier B is submultiplied by the multiplier A, as the skin element B is the same as the element of the multiplier A. It is indicated by the sign of inclusion: B ⊂ A.

From the submultiple, be it a multiplier, you can see your submultiple. For example, among the students of the class you can see a lot of girls, and among the girls you can see a lot. Todi can be written like this:

Ce means that the impersonal C is included in B, and B is included in A.

If the multipliers are marked with stakes, then in the middle of the stake A will be the stake B, and in the middle of the stake C. Similar little ones are called Euler-Venn diagrams.

As two multipliers are equal, then for them the consonances A ⊂ B and B ⊂ A are counted.

It is already established that B ⊂ A, and if any element x lies in B (x ∈ B), then it means that x ∈ A. element lie down B. It could be wrong.

Bezlich- Sukupnіst be-yakikh ob'єktіv. Bagato signify great letters of the Latin alphabet - vіd A before Z.

The main numerical multipliers: the numberless natural numbers and the numberless whole numbers, which are always denoted by the same letters:

N- impersonal natural numbers

Z- impersonal numbers

Multiple element- tse be-yaky object, scho to enter the warehouse multiplier. The belonging of the object to the multiplier is indicated by the additional sign ∈ . Record

it reads like this: 5 to lay a lot Z or 5 is the element of the multiplier Z .

Anonymous to share in the endings and unskinned. Kintseva bezlich- a multiplicity to avenge the song (Kintsev) a lot of elements. Neskіchenna impersonal- impersonal, what to avenge impersonal elements. Up to innumerable multiplies it is possible to include an infinite number of natural and whole numbers.

For the designation, many victorious arches are curly arches, in which the elements are perked through. For example, recording

L = {2, 4, 6, 8}

means that you are faceless L is added up with four pairs of numbers.

The term impersonal gets used independently, depending on how many elements it is necessary to avenge. Many do not avenge the same element are called empty.

Submultiple

Submultiple- ce impersonal, all the elements of such a part of another multiplier.

Visually demonstrate the multiplier and submultiplier that you can enter before a new one, you can get help kіl Euler. Cola Euler - ce geometric schemes that help to visualize the visibility of various objects, in our opinion, many.

Let's look at two multipliers:

L= (2, 4, 6, 8) and M = {2, 4, 6, 8, 10, 12}

Leather element multiple L lie down and faceless M father, faceless L M. Such a multiplication of multipliers is indicated by the sign ⊂ :

L⊂M

Record L⊂M read like this: faceless Lє multiplicity of impersonality M .

Impersonal, which are formed from the quiet elements themselves, independently in their order, are called equal it is denoted by the sign = .

Let's look at two multipliers:

L= (2, 4, 6) and M = {4, 6, 2}

So, as many insults are added up from one and the same elements, then L = M.

Peretin and the union of many

Peretin two sets- tsukupnіst elementіv, scho belong to the skin z tsikh multiplies, tobto їhnya zagalna part. Peretin is denoted by the sign ∩.

For example, like

L= (1, 3, 7, 11) and M= (3, 11, 17, 19), then L∩M = {3, 11}.

Record L∩M read like this: peretin multiplier Lі M .

Whose butt is screaming, what a multiplicity is called a multiplicity, so that only those elements can be avenged, as they grow in all the multiplies that are intermingled.

Combine two sets It is called impersonal, which is to avenge all the elements of the past multiples in a single instance, so that if one and the same element is summed up in both multiples, then in a new multiple, this element will be included only once. The union means the sign ∪ .

For example, like

L= (1, 3, 7, 11) and M = {3, 11, 17, 19},

then L∪M = {1, 3, 7, 11, 17, 19}.

Record L∪M it reads like this: the union of many Lі M .

When combining equal multiples, combining more than cutaneous multiples:

yakscho L = M, then L∪M = Lі L∪M = M.