What is so powerful multiplier. Understand the multiplier. Multiply. Elements of combinatorics: Permutations

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 lies in the fact that it is a firmness (powerfulness) that is fair for a certain cob natural number n 0 and even though it is true for a sufficient natural number n = k ≥ n 0, it can be proved true for the number k + 1, then this power is valid 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.

Bezlich BUT name multiplied faceless S(otherwise the multiplier is 5), as the skin element of the multiplier BUTє an element of a multiplication S. Designation: Іс5.

Viraz AqS also read: "BUT included up to 5", "BUT avenge at 5", "S take revenge A", "A part S. Sign z name enable symbol.

Let's write down the appointment symbolically:

Z vyznachennya vyplivaє: bezlіch BUTє multiplied in S even and only once, if from the proposition ( heA) following proposition (xes).

Let's stop it until AczS. Behind the laws of logic may be:

Father, the proposition "Many BUT not included until S» is equivalent to the proposition «Isnuє element of the multiplier BUT, which does not lie at S.

Two multiples BUTі At Formally, it is possible to connect with inclusion signs in two ways: A(z.Bі VsA. Kozhen іz tsih virazіv vyznaє proposition, yak can be true or hibnoy. Other included B (also write A^B) hundred of the first is called the return. Do not start from the justice of one of the inclusions, the truth of the other inclusion is vibrated.

Stock 6.2.1. May include (-2; 2) s (-2; 0; 1; 2), so that the numbers (-2) and 2 are the elements of the multiplier (-2; 0; 1; 2). However, (-2; 0; 1; 2) is not included in (-2; 2), so, for example, 0r (-2; 2).

Stock 6.2.2. Come on BUT - impersonal usikh rhombuses, AT - the multiplicity of all squares.

BUT ? AT, oskelki is a rhombus, but not a square.

PER, oskіlki be-like a square є rhombus (which screams from the designation of these figures).

In other words, the facelessness of all squares is the submultiple of the facelessness of all rhombuses.

Stock 6.2.3. (X| *:12)s(* | dg:3), shards *:12=>*:3 (ground yourself). However, turning it on is wrong, because x:3ph":2(Introduce a counter-butt).

Z vznachennya vyplivaє, that the skin is impersonal є pіdnіzhinoyu itself.

Take a substitute BUT empty lot. Todi hardening 0qS equal to V * (Xe0 xeS). Oskіlki posilka іmplikatsії zavzhdi khibna, then whether there is any object d; the implication gains a reference value. Otzhe, firmness 0qS is true. Father, an empty multiplicity is a submultiple, be it a multiplier.

Visnovok: be it a non-empty multiplier, make two submultiples - the multiplicity itself is empty. They are called trivial submultiples. The very impersonal is also called an unscrupulous submultiple.

Pіdnіzhina v S called let's take it, yakscho won't run out of S. Record AaS means that BUTє vlasniy pіdnіzhinoy S:

The symbol of the severe inclusion is the sign.

We may have two extensions: the inclusion of an element to a multiplier (which is indicated by the sign e) and the inclusion of multiples (which is indicated by the sign z). Zagalom tse different signs. For example, (2) s (2.3), ale (2) * g (2.3). However, sometimes you can put insulting signs in multiples.

Stock 6.2.4. Bezlich A =(2) є multiplier element At= (1,2, (2)). With whom BUTє pіdnіzhina impersonal At so vaga elements multiply BUT lie by At(at BUTє only one element - the number 2, like to lie in U).

Also, (2)e(1,2,(2) and (2)c(1,2,(2)).

Stock 6.2.5. Let's look at the flat a i straight / that lie on this plane. How to look straight as an element of the plane, it is customary to write lea. How can you understand a straight line, like an impersonal point, that this straight line should lie, then impersonality will be a multiplicity of impersonality of all points of the plane. Todi can be written /sa.

Let's go straight and turn it on AqB ta B Do you want me to X vykonuyutsya implications xEL -> xEBі xgB->xeL, which is the same as for all. heL then and only then, if heV. Tse means that the faceless BUTі At avoid:

It is simple to use it to lay the foundation for the method of proving the equality of multiples, called by the method of suspended inclusion: in order to bring the multipliers A and B equal, it is necessary to bring straight and reverse the inclusion of multiples.

In fact, this idea was demonstrated in application 6.1.3, the shards are directly included A means that the predicate P(x), ask impersonal BUT follow predicate Q(x), what is impersonal At, and the turn-on means that Q(x) next P(x). Let's look at one more example.

Stock 6.2.6. Take the impersonal:

A = {2P | neZ)- impersonal of all guy numbers,

B - (xx \u003d a + b, de aі b- unpaired numbers) - the multiplier of all numbers, the skin of which is the sum of the deyaky unpaired numbers.

Let us know that A = St.

Let's show the fairness of the inclusion A^B. Come on hea, maybe x = 2w = (2/f-l)+1, so X we can imagine having a sum of two unpaired numbers. To mean, heV.

Correctly also turned on VsA. Right, let's not heV. Todi x =(2/7+1)+(2A+1) = 2(/;+A"+1) = 2t. To mean X - guy number, to heA.

Offended inclusion brought. So, multiply BUTі At equal.

Right. Bring what you multiply (2/7-1 ne Z) and (2/7+1 | not Z) equals, so that insults signify the absence of unpaired numbers.

Stock 6.2.7. Respectfully, scho faceless A =(2n-1 | // eN) ta At- (2/7+1 | /7 € N) ns equal, shards IgA, ale 1 &AT. To that, the faceless of us unpaired positive numbers ask only stupid people L. When you turn on L^B true.

Let it be given to the impersonal S. Family of all submultiples S called boolean impersonal S(otherwise step of the multiplier S) and is indicated (5) otherwise 2s.

For appointment B(5) = (X| AfcS).

It is clear that 0eB(5) and SeB(S) for whatever multiplier S.

Stock 6.2.8. Come on S= (1,2,3). We know bulsan qієї multiplier.

It is respectful that the elements of the bulean are multiplied.

The term “multiple steps” and the second recognition are motivated by the fact that if we can Kіntsev //-elemental multiplicity, then the number of elements in the yogo bulean will be a more advanced level 2. Well, the formula will be examined, which allows you to know the number of submultiples in the //-elemental multiplier, so that the number of elements is fixed.

In the deakіy literature it is known that it means quite a lot.

Two multiples of A and B are equal, as if the stench is added up from the same and the same elements.

According to this principle, it is obvious that for any two different multitudes, there must always be a certain object, which is an element of one of them and not an element of the other. Shards of empty wastefulness avenge the elements, all the stench commemorate that empty emptyness is one and the same.

Multiply. The meaning of the equalness of the plurals can be formulated in another way, the vicarious understanding of the plural.

Appointment. The multiplier A is called the submultiplier of the multiplier B, so that element A is element B.

Last 1. Obviously,
for the multiplier A, because skin element A є element h A.

Last 2. For be it a multiplier A,
If the empty multiplier was not filled with A, then in the empty submultiple they used elements so that A would not lie.

Yakscho
, then write
, i
, then A is a good subset of B.

The understanding of the multiplication of multiplies allows you to easily formalize the understanding of the equality of two multiplies.

Firmness. For whatever A and B

The logical equivalence, which is shown by virase (1.1), is the main way to prove the equality of two multiplies.

Respect . Inclusion  May have a number of obvious powers:

(Reflexivity);

(Transitivity).

For any multiplier X, you can designate a special multiplier of all submultiples of X, as it is called Boolean ℬ
, Yake includes in itself the impersonal X, all the yogo increase and empty the impersonality
.

butt. Come on
- Tse impersonal, which is composed of three elements. Todi bulean ℬ (X) ce impersonal:

Vlasniy pіdnіzhina ℬ (X) є next multiplier:

(a), (b), (c), (a, b), (b, c), (a, c).

In a wild mood, if a multiplier of X to revenge n elements, then a multiplicity of yogo submultiples ℬ (X) adds up to elements.

Operations on multiplies.

Come on U is a universal multiplier,
. Then, for the multiplies X,Y, one can assign the operations
.

Appointment . The combination of multiples X and Y is called impersonal
, What are the elements, what should be included in one of the multiples (X or Y):

Rice. 1.1- Combination of multiples Rice. 1.2- Peretin multiples

Appointment . Peretin of multiples X and Y is called impersonal
, What are the elements that are included in X and Y at the same time:

Appointment . The retail of many X and Y is called impersonal
, What elements are added up, what should be included up to a multiplier of X, but not included up to a multiple of Y:

Rice. 1.3- Retail of multiples
Rice. 1.4– symmetrical

difference of multiples

Appointment . The symmetrical cost of two sets X and Y is called impersonal
, What is added up from the elements of the multiplier X and the elements of the multiplier of Y, for a few elements, which are the most important for both multipliers:

Appointment . For be-like a multiplier
add to the impersonal before U is called taka impersonal , what:

Rice. 1.5– Addendum multiply X to U

On fig. 1.1  1.5 Represented by Venn diagrams, as a direct demonstration of the results of operations
.

Additional multipliers are indicated
. Operations
tied to each other by de Morgan's laws:

, (1.7)

. (1.8)

The justice of de Morgan's laws is easily swayed independently.

In Table 1.1, the main power of operations over multiples is indicated.

Table 1.1

	The power of operations	Consolidation, retin, addition
	commutativity	,
	associativity
	distributiveness
	idemopotency	, , , , ,
	de Morgan's theorem	,
	involution

Operations of joining that webbing can be done. Come on
- anonymous indexes,
- Family of submultipliers of X.

Appointment. Family of submultiples
multiplier X, for those
, called rozbittyam multiplier X, so that two minds are counted:

Appointment. Family of submultiples
multiplier X is called pokrittyam multiples of X, so:
.

Appointment. The class K submultiples U is called an algebra, which means:

1.
;

2. because
screaming what
;

3. from what
screaming what
.

butt. Come on
same class
I approve algebra.

Appointment. Class F -algebra, like:

1.
;

2. because
next
;

3. from what
,
screaming what
.

butt. The impersonality of all subdivisions U make -algebra, tobto. ℬ (U)- -algebra.

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

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

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 $\subset$ 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 $B \supset A$ .

The impersonality of all submultiples $A$ be appointed $\mathcal(P)(A)$ 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 $A\subset B$ і $A \ne B$ .

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 $A \ne\varnothing$ .

Apply

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

power

The establishment of a multiplicity of maє tsіlu low authorities.

Remembrance of the multiplier statutes of private order :
- Remembrance day reflexively : $B\subset B$
- Remembrance day antisymmetric : $(A \subset B \; \and \; B \subset A) \Leftrightarrow (A = B)$
- Remembrance day transitively : $(A \subset B \;\and \; B \subset C) \Rightarrow (A \subset C)$
Empty facelessє a multiplicity be-anything else, to him є the smallest multiplicity of a multiplier of a multiplier: $\varnothing\subset B$
For some kind of two multiplies Aі B offensive hardness equivalent:
- $A\subsetB.$
- $A\capB=A.$
- $A \cup B = B.$
- $B^(\complement) \subset A^(\complement).$

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 $2^n$ submultiple (including empty). Sob to perekonatisya in tsoma, dosit pomititi 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 $k\le n$ elements, then their number is expressed binomial coefficient $\textstyle\binom(n)(k)$ . 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 $\textstyle\frac(n(n-1)\dots(n-k+1))(k=\binom(n)(k)!}$ 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, gentlemen, no ... don’t think ... I’ll be more reasonable, you 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 town, stars and from that side of the bridge.