Repair

Golovna

De Morgan's laws are logical rules, established by the Scottish mathematician Augustus de Morgan, which relate pairs of logical operations to a logical sequence.

Augustus de Morgan respected that classical logic has the following fair relationships:

not (A and B) = (not A) or (not B)

not (A or B) = (not A) and (not B)

In a more basic form for us, the relationship data can be written in this form: De Morgan's laws can be formulated as follows:

I de Morgan's law: The intersection of disjunctions of two simple ones is equal to the conjunction of the intersection of these two.

II de Morgan's law:

The intersection of the conjunctions of two simple definitions is equal to the disjunction of the intersection of these definitions. Let's take a look at the construction of de Morgan's laws on specific butts.

butt 1.

Rework the formula so that there is no overlap of folding visors.

,

In accordance with De Morgan's first law, we can reject:

.

Before the conjunction of simple ones is crossed, another de Morgan law is established, it is rejected:

in this order:

As a result, even more force was removed, for those who do not have a list of warehouse connections, the restrictions will be reduced to simple results.

.

You can verify the validity of a decision using an additional truth table.

Table 1.

B/\C A/B/\C As per the table, the more logical derivation and the more logical derivation, based on de Morgan’s laws, are equivalent.

This is due to the fact that we removed new sets of values ​​from the truth tables.

The considered operations on the anonymous are ordered by certain laws that represent the elementary laws of the algebra of numbers.

10'. Poretsky's Law

11. Law of involution (subsequent addition)

The laws of algebra of multiplicities are based on the operation wedge () and ob'edannaya () are ordered by the principle of duality: as in any law, all wedge signs are replaced by ob'edannaya signs, and all obedannaya signs are replaced by wedge signs, universe (U) replace with the sign of the empty multiplicity (Ø), and the sign of the empty with the sign of the universe, then the new true identity is eliminated.

For example (on this principle), next, etc.

3.1.

Verification of the truth of identity for additional Euler-Venn diagrams

All laws of the algebra of multiplicities can be clearly understood and explained using Euler-Venn diagrams. For whom is it necessary:

Draw a line across the diagram and shade all the multiples that are on the left side of the line. Place another diagram and create the same one for the right side of equality.

Draw a line across the diagram and shade all the multiples that are on the left side of the line. This sameness is true only if the same area is shaded in both diagrams.

Respect 3.1.

Two stakes, which shift, divide everything universally impersonal in several areas (div. Fig. 3.1)

Respect 3.2. Three stakes, which move, divide all the universal impersonality in all areas (div. Fig. 3.2):

When recording the minds of different applications, the following meanings are often used:

 - with…following…; 3 .2  - this and that, if… .

Zavdannya 3.1

.

When recording the minds of different applications, the following meanings are often used:

Forgive the virazis of multiply algebra: Decision.

When recording the minds of different applications, the following meanings are often used:

Zavdannya

.

Bring the sameness:
(AB)\B = A\B; A(BC) = A\(A\B)(A\C). Zavdannya 3.3
. A(BC) = A\(A\B)(A\C).
Bring the relationship to fruition in two ways: using an additional diagram and using an additional equalization of multiplicities. A(BC) = A\(A\B)(A\C). 2. Proof of additional significance of the equality of multiplicities. A(BC) = A\(A\B)(A\C).
For the meanings, the multiplicities of X and Y are equal, since at the same time the relationship is determined: XY and YX. A(BC) = A\(A\B)(A\C). Let's show you what A(BC) = A\(A\B)(A\C).. A(BC) = A\(A\B)(A\C). Let's go A(BC) = A\(A\B)(A\C). X
- additional element of the multiplicity A(BC) = A\(A\B)(A\C)., then
.
Tse means that
U that
.

The star is screaming
(AB)\B = A\B;
Or A(BC) = A\(A\B)(A\C).B. A(BC) = A\(A\B)(A\C). 2. Proof of additional significance of the equality of multiplicities. A(BC) = A\(A\B)(A\C). Yakshcho A(BC) = A\(A\B)(A\C).Ah, then then
- additional element of the multiplicity
Ā, which means A(BC) = A\(A\B)(A\C). 2. Proof of additional significance of the equality of multiplicities. A(BC) = A\(A\B)(A\C).. A(BC) = A\(A\B)(A\C). Well
.
The star vibrates that the skin element multiplies
.
.

є also an element of the multiplier
To mean,

, what needed to be brought out.

A(BC) = (AB)(AC);

1. Proof with additional diagrams: A(BC) = A\(A\B)(A\C). Let's go A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C). Todi A(BC) = A\(A\B)(A\C).And that A(BC) = A\(A\B)(A\C).BC. A(BC) = A\(A\B)(A\C). Yakshcho A(BC) = A\(A\B)(A\C).B, then A(BC) = A\(A\B)(A\C).AB, why don’t you say too much about what was said, but then, A(BC) = A\(A\B)(A\C).(AB)(AC).

Well A(BC) = A\(A\B)(A\C).С, then A(BC) = A\(A\B)(A\C).AC. A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C). Otje, A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C).(AB)(AC). A(BC) = A\(A\B)(A\C). Well, it’s clear that A(BC) (AB)(AC. A(BC) = A\(A\B)(A\C). Let's go now A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C). (AB)(AC). A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C).(AB)(AC). A(BC) = A\(A\B)(A\C). Yakshcho

AB, then
B.

Zavdannya

The following is what follows: A(BC) = A\(A\B)(A\C).ВС, then
.

A(BC). A(BC) = A\(A\B)(A\C). Well
.

АС, then A(BC) = A\(A\B)(A\C).And that A(BC) = A\(A\B)(A\C).S. A(BC) = A\(A\B)(A\C). The star is screaming
.

A(BC).
In this order, (AB)(AC) A(BC).
Otzhe, A(BC) = (AB)(AC).

Well
і A(BC) = A\(A\B)(A\C). What needed to be brought up. A(BC) = A\(A\B)(A\C). For the proof of sufficiency, we rejected that AB =.

АС, then A(BC) = A\(A\B)(A\C). Obviously, the relationship has been brought to an end. A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C). Behind the proof there was a most extreme twist.
However, there are a number of possible options for everyday diagrams. A(BC) = A\(A\B)(A\C). For example, the fall of equality AB=C or

A(BC) = (AB)(AC);

Zavdannya

, loss of empty multipliers and so on. A(BC) = A\(A\B)(A\C). Obviously, it is important to consider all possible options for insurance.
Therefore, it is important that the proof of the relationship with other diagrams is not always correct. A(BC) = A\(A\B)(A\C). Necessity. A(BC) = A\(A\B)(A\C). Let ABC that element A.
Let us show that in this case the element of multiplicity A will also be an element of multiply

Let's look at two types: Or

When recording the minds of different applications, the following meanings are often used:

Yakshcho

ABC, then x xС, і, as whose inheritance, A(BC) = A\(A\B)(A\C). Well A(BC) = A\(A\B)(A\C)., then th A(BC) = A\(A\B)(A\C).. A(BC) = A\(A\B)(A\C). The need has been brought to light. A(BC) = A\(A\B)(A\C).AB.

Let's show what element

will also be an element of the multiplier C.

AB, then

B. Oskolki, to mean

S. x xС, і, as whose inheritance, x The sufficiency has been communicated. A(BC) = A\(A\B)(A\C)., then th A(BC) = A\(A\B)(A\C).A(BC). A(BC) = A\(A\B)(A\C). Well A(BC) = A\(A\B)(A\C). Let AV.

Let's take a look at the element

1. B (or

2. ).

Zavdannya 3.1

Similar:

A (or

Ā).

3. Then every element is multiplied

AX = A;

(vіdpovіd );

Formulas and laws of logic At the introductory lesson dedicated to fundamentals of mathematical logic We have become familiar with the basic concepts of this branch of mathematics, and now the topic is naturally going to be continued. In addition to the new theoretical, or rather not theoretical - but behind-the-scenes material, practical tasks are waiting for us, and because you came to this page from a joke and/or are poorly oriented to the material, then, please, pass Follow the instructions of the fortune-tellers and start from the beginning statti. Besides, for practice we need 5 truth table logical operations.

like me

I highly recommend it
rewrite by hand
DO NOT memorize, DO NOT unlearn, but once again make sense of yourself and copy it onto paper with your best hand - so that the stench will be in front of your eyes:
- Table NOT;
- Table I;

- ABO table; - Implication table;- Equivalence table.

This is very important. In principle, they would have to be numbered manually: "Table 1", "Table 2" etc., but I have repeatedly supported this approach - as it seems, in one case the table will appear first, and in another - a hundred first.

To that vikorist we call it “natural”.

Continued:
.

In fact, you already know the concepts of the logical formula.

I’ll set it up in a standard way, but wait until then appointment formulas Vislovlyuvan algebras are called: 1) be some elementary (simple) understanding; 2) if i are formulas, then the formulas also express the form There are no other similar formulas.

A formula is a logical operation, for example, a logical multiplication. To return respect to another point - he allows recursive

to “create” the formula as many times as possible. Oskolki- Formulas, then - the same formula;


fragments i – formulas, then - The same formula, etc. Be-yake elementary vislovlyuvannya (I'll let you know again soon with the appointments) You can enter the formula more than once. Formula.

Not For example, writing - and here there is an obvious analogy with the “schemes of algebra”, which is not clear - it is necessary to add or multiply numbers. The logical formula can be seen as

logical function all possible combinations Truths and lies that can be accepted by elementary reasoning (arguments).

This formula includes two formulas, and it doesn’t matter what such combinations are. “At the output”, the logical values ​​of the entire formula (function) are determined.:

It is necessary to say that the “exit” here is “in one time,” but the formula is logical and complex.
And in such “difficult situations” it is necessary to stay alert
order of logical operations
- We are faced with a list of restrictions;
– in a friend’s chergu – conjunction;

- Then - disjunction;

- Then implication;
- And, agreed, the lowest priority is equivalence.

For example, a record is transferred that requires a logical multiplication, and then a logical addition: . Just like in “primary” algebra – “from the beginning it is multiplyable and then it is foldable.” The order of actions can be changed in the primary way - with arms:– here we are faced with a disjunction and then a “stronger” operation. Singingly, everyone will understand, but against every burner

: і tse two differences formulate!:

(both formally and informally) Let's create a truth table for the formula. This formula includes two elementary conditions and “at the input” we need to sort out all possible combinations of ones and zeros.

To avoid misunderstandings and misunderstandings, be sure to overinsure the combinations. no problem with this order(which I am, in my opinion, de facto vikorist from the very beginning) The formula includes two logical operations, and according to their priority, you first need to check prohibited vislovlyuvannya.:

Well, let’s close the “ne” section – the ones are converted into zeros, and the zeros into ones:

On the other hand, we marvel at the stanza and zastovoem before them ABO operation Be-yake elementary vislovlyuvannya .:

Getting ahead of myself a little, I will say that disjunction is permutable

(i – not the same) , And therefore the conditions can be analyzed in the primary order - from left to right. and independently analyze equally strong statements.

Let me formulate in the salon: two formulas are called equal (identical) how they take on new meanings regardless of the set of values ​​that are included in these formulas (elementary vislovlyuvan). So it seems that“Formulas are equivalent if their truth tables are consistent”

But this phrase doesn’t really suit me.

Zavdannya 1

Compile the truth table for the formula and check the validity of the identity you know.

Let us repeat the order of the task once again:

1) Since the formula includes two variables, then there will be 4 possible sets of zeros and ones. We write them down in a deliberate order. 2) The implications of “weak” for the conjunction, otherwise they will be spread out in the arms. Of course, it’s important to understand this kind of application:“if a one is followed by a zero, then we put a zero, in all other cases we put a one”

. Next, we repeat the standard for the implication, and therefore,.

Respect! - Stovpts and traces analyze “right-handed to left”! .

3) And at the final stage, the pouch section is refilled.

And here it’s easy to blurt it out like this:

“if there are two ones in the columns, then we put one, in all other cases we put zero”

I, we find it, check it from the truth table equivalents


The basic equivalences of algebra are explained We only got to know the two of them, but the others on the right, obviously, cannot be separated. There are a lot of similarities and I will go over the most important ones and find them: Commutativity of conjunction and commutativity of disjunction Commutativity- This is permutability: Know the rules from 1st grade:“When the multipliers (addanks) are rearranged, the amount (sum) does not change” ..

Although, despite everything, there is an impression of the elementary nature of this power, it is fair to say that it will not be forever, strictly speaking, non-commutatively multiplication matrixі (In the zagalny form they cannot be rearranged), A

vector additional vectors

– anticommutative


(rearrangement of vectors causes a change in sign)

Please note that in other people it is incorrect to talk about “opening the arms”, but in the senses there is a “fiction” - they can be taken away: because

Multiplication is a powerful operation. And again, it would seem that “banal” powers do not apply to all algebraic systems, and, moreover, they require proof(we’ll talk about them soon)

.

Before speaking, another distributive law is unfair in our “primary” algebra.


And to tell the truth:

Law of idemopotency


What about robiti, latin.

This is exactly the principle of a healthy psyche: “I and I are the same”, “I and I are the same” =)

There are several similar identities:
...hmmm, now I’m completely finished... so I can become a Doctor of Philosophy tomorrow =)

The law of subdivision

Well, here it’s already asking for an example from the Russian language - it’s wonderful to know that the two parts “not” mean “so”.

And in order to strengthen the emotional barrier, three “don’ts” are often used:

- the final piece of evidence came out!

Laws poglinannya - “What about you, lad?”=) On the right side, the arms can be lowered. і – De Morgan's Laws It’s acceptable that suvory vikladach (Whose name you also know :)) To return respect to another point - he allows put ispit, because – Student vidpov for 1st exam (Whose name you also know :)) To return respect to another point - he allows Student vіdpovіv on another student . Todi vyslovlyuvannaya about those who.

Student

Having passed the exam , will be equal to the firmament -:

Power up for 1st power supply

or else for 2nd meal As it was meant above all, the equivalences contribute to the proof, which is usually done with the help of a truth table.

In fact, we have already achieved the equalities that express the implication and equivalence, and now the time has come to consolidate the technique of this highest task.

Let's bring the sameness.

Next, we create a truth table for the right side. , will be equal to the firmament -:

Here everything is clear - we are first carrying out the “strongest” restrictions, then we are stagnating until the end

The results were consistent, and thus the identity was achieved. If equality can be given to the eye same true formula . Tse means that

FOR ANY output set of zeros and ones

"on the way out" there is strictly one person coming out.

And this has a very simple explanation: the fragments of the truth table are saved, then, obviously, they are equivalent.

Or, more compactly:

Zavdannya 2 Bring the following equalities: b)

A short solution to the end of the lesson.

Don't be lazy!

Don't just build truth tables, but also clearly formulate symbols. As I have uncharitably designated, simple speeches can cost the stranger dearly! Be-yake elementary vislovlyuvannya Let's continue to get acquainted with the laws of logic!.

So, absolutely correct - we already practice with them:

Truth at clearly , called formulate symbols. also the true formula Be-yake elementary vislovlyuvannya law of logic.

Looking at the previously discussed transition from equivalence to a completely true formula, all of the above are similarities and laws of logic.
The formula for how meaning emerges

Nonsense

be any set of values ​​that must be included before it also with the same formula.

protirichchyam

The branded butt rubs off from the ancient Greeks:

- Everyday expression can be true and false at the same time.

The proof is trivial:

“At the output”, all zeros are removed, so the formula works totozhna hibna However, whatever the supernaturalness, this is also the law of logic, literally:

It is impossible to cover such a great topic in one single article, so I will limit myself to just a few laws:

Law of the excluded third - In classical logic there is no third choice.“Buti chi don’t buti” is the axis of the diet. Fold the truth sign yourself and check what it is.

also true formula. Law of contraposition This law was actively discussed when we discussed the essence necessary mind, guess:

If at the hour I find there is dampness on the street, then it comes out because if it is dry on the street, then there was definitely no rain. It also follows from this law that it is fair straight theorem:

, then the affirmation, as some people call it, will certainly be true protilegone theorem. formula.і It's true gateway It also follows from this law that it is fairі theorem theorem.

For the “adult” formulation of the Pythagorean theorem, all 4 “directions” are true.

Law of silogism Same classic genre:.

“All oak trees are trees, all trees are trees, and all oaks are trees.”

Well, here again I would like to introduce the formalism of mathematical logic: since our sage Vikladach thinks that every Student is an oak tree, then from a formal point of view, the Student is insanely tall =) ... if you want to think about it, then maybe even from an informal point of view = )

Let's create a truth table for the formula. This is consistent with the priority of logical operations, as determined by the algorithm: 1) by implication.

Once it seems, you can immediately conclude the 3rd implication, and then it will be easier , will be equal to the firmament -;

(and acceptable!)

get married for a little while; 2) until stovptsіv stagnation

3) the axis is now flexible;


4) and at the final end the stasis is implied by the implication to the conclusions
To mean,

ta .

Don’t hesitate to control the process with your middle and middle fingers :))


The rest of the story, I guess, everything made sense without comment:

Zavdannya 3

Know that the law of logic is the following formula:

A short solution to the end of the lesson.



So, and without forgetting a little, let’s think about over-inflating the output sets of zeros and ones in the same order that is a proof of the law of silogism. The rows can be rearranged, but it will be better to arrange the animal according to my decisions. Reworking logical formulas

The principle of its “logical” significance and equivalence is widely used for the re-interpretation and simplification of formulas. Roughly speaking, one part of the similarity can be replaced by another.:


So, for example, if in a logical formula you already know a fragment, then following the law of idempotency, the replacement of it can (and should) simply be written down. :


If you are vibing, then ask for the record before the law.


And so on.


In addition, there is another important point: the sameness is true not only for elementary formulas, but also for advanced formulas.


Say out loud the logical phrase “with oak, tree, deed,” and you will understand that after the rearrangement of the implication, the replacement of the law of anitrochy has not changed.

If only the formula had become original.

As training, we ask for a formula.

Start with what? First of all, let’s take a look at the order of actions: here the cross-sections are stacked up to a whole arch, which is “fastened” to the elements by a “little weak” conjunction. In fact, there is a logical combination of two multipliers: .

With two operations that have been omitted, the lowest priority is the implication, and therefore the whole formula has the following structure: . As a rule, on the first step(s) the equivalents and implications are reduced

(what a stench it is)


and reduce the formula to three basic logical operations.

What can I say?

Logical.

(1) Vikoristic identity

I, we find it, check it from the truth table

.

And to our vipadka.

Then begin the “disassembly” of the arms.

First everything is decided, then comments.

So that the oil of oil does not work out, the icons of “extreme” zeal will be used:

(2) To the outer arches, de Morgan's law is stagnated, de.

Association

x 1 (x 2 x 3) = (x 1 x 2) x 3;

x 1 Ú (x 2 Ú x 3) = (x 1 Ú x 2) Ú x 3 .

x 1 x 2 = x 2 x 1

x 1 Ú x 2 = x 2 Ú x 1

Distributivity of conjunction before disjunction





x 1 (x 2 Ú x 3) = x 1 x 2 Ú x 1 x 3 .

Distributivity of disjunction before conjunction

x 1 Ú(x 2 × x 3) = (x 1 Úx 2) × (x 1 Úx 3).

*

Idemopotency (tautology) Podviyne interlocked .

The power of constants x & 1 = x; .

(Laws of universal multiplicity)

x & 0 = 0; The disjunction of two elementary conjunctions, one of which is a storage unit, can be replaced by a conjunction that has fewer operands .

The rule for the creation of elementary sums follows from a different kind of law and the law of zero multiplicity: The conjunction of two elementary disjunctions, one of which is a storage unit, can be replaced by an elementary disjunction, which has fewer operands.

Rule of the throat

This rule means that the gate is glued together.

The rule for the development of an elementary creation into a logical sum of elementary creations of a higher rank (between up to r = n, then up to the constituent of one, as will be discussed below) arises from the laws of universal multiplicity, which is similar to the law of the first kind and takes place in three stages:

At an elementary level, which flares up, r is introduced into the containers of spp. n-r one, de n-rank constitution of one;

The skin unit is replaced by a logical sum of actions, but the output elementary creation does not have the following changes: x i v `x i = 1;

The opening of all the arms is carried out on the basis of the diversified law of the first kind, which leads to the opening of the output elementary addition of rank r in a logical sum of 2 n-r constituents of one.

The rule of the larynx of an elementary creation is modified to minimize the function of logic algebra (FAL).

The rule for the development of an elementary sum of rank r to the formation of elementary sums of rank n (constituent of zero) is consistent with the laws of the zero multiplicity (6) and the corresponding law of another kind (14) and is carried out in three stages:

At the sumu rank, it’s getting hot, r like dodanki are introduced n-r zeros;

The zero seems to be a logical solution to the action, but the output amount does not change and is listed: x i·` x i = 0;

Viraz, which is superior, is transformed on the basis of a separate law of a different kind (14) in such a way that the output sum of rank r flares up into a logical addition 2 n-r constituent of zero.

16. Concept of a new system.

Applications of external systems (with confirmation) Viznachennya.

An impersonal function of the algebra of logic A is called a complete system (P2), since any function of the algebra of logic can be expressed by a formula over A. System of functions A = ( f 1 , f 1 ,..., f m ), which is again called.

basis A minimal basis is such a basis for which one function is required for any particular function. f 1 , which creates this basis, recreates the system with a function(f 1, f 1, ..., f m)

u nepovnu. Theorem.

Finished.

Since the function of the algebra of logic f is expressed as identical zero, then f is expressed in what appears to be a completely disjunctive normal form, which includes disjunction, conjunction and concurrence. If f ≡ 0, then f = x & x.

The theorem has been proven.

Lemma.

Since system A is a complete system, and if any function of system A can be expressed by a formula over another system B, then B is also a constant system.

Finished.

Let’s take a look at the logic algebra function f (x 1, …, x n) and two systems of functions: A = (g 1, g 2, …) and B = (h 1, h 2, …).

Looking at the fact that the system A is complete, the function f can be expressed by looking at the formula above it:

u nepovnu. f (x 1 , …, x n) = ℑ

de g i = ℜ i

then the function f is represented in the view

f (x 1 , …, x n)=ℑ[ℜ1,ℜ2,...]

Otherwise, apparently, it can be represented by a formula over B. By going through all the functions of algebra logic in this way, it is clear that system B is also complete.

The lemma is complete.

Advanced systems and advanced P 2:

4) (&, ⊕, 1) Zhegalkin base.

Finished.

1) It is clear (Theorem 3) that the system A = (&, V,) is complete. Let us show that the complete system B = (V, . It is valid, from de Morgan’s law (x& y) = (x ∨ y) it is derivable that x & y = (x ∨ y), then the conjunction is expressed through the disjunction and listed, and all functions of system A are expressed by formulas over system B. System B is similar to the same..

2) Similar to point 1: (x ∨ y) = x & y ⇔ x ∨ y = (x & y) and from line 2 the truth of the confirmation of point 2 is evident.

1. 3) x |

y=(x&y), x |

2. x = x;

x & y = (x | y) = (x | y) |

3. (x | y) ta zgidno lemi 2 system povna.

4) x = x ⊕1 i zgіdno with lemoy 2 system is full.

4. The theorem has been proven.

17. Zhegalkin algebra.
The power of operations is repeated The absence of Boolean functions specified in the Zhegalkin basis S4=(⊕,&,1) is called

Zhegalkin algebra

Main characteristics.

commutability

h1⊕h2=h2⊕h1 h1&h2=h2&h1 association h1⊕(h2⊕h3)=(h1⊕h2)⊕h3 h1&(h2&h3)=(h1&h2)&h3

distributivity

h1&(h2⊕h3)=(h1&h2)⊕(h1&h3)

power of constants

For example, f=x⊕yz⊕xyz and f 1 =1⊕x⊕y⊕z are polynomials, and the other is a linear function.

Theorem.

Each Boolean function is represented in the form of a Zhegalkin polynomial by a single number.

Let us introduce the basic methods of generating Zhegalkin polynomials for a given function.

1. Method of insignificant coefficients.

Let P(x 1 ,x 2 ... x n) - Zhegalkin polynomial that implements the given function f(x 1 ,x 2 ... x n). Let's write it down in sight P=c 0 ⊕c 1 x 1 ⊕c 2 x 2 ⊕ ... ⊕c n x n ⊕c 12 x 1 x 2 ⊕ ... ⊕c 12 ... n x 1 x 2 ... x n

We know the coefficient Ck. For which it is necessary to change x 1 x 2 ... x n the values ​​of each row of the truth table. The result is a system with 2 n equals with 2 n unknowns, which may

One decision.

Having verified this, we know the coefficients of the polynomial P(X1, X2...Xn)..
2. Method based on the transformation of formulas over anonymous links (&).

There will be a formula

F

over the unlinked links (,&), which implements this function f(X 1 ,X 2 ... X n).

Then we replace here the form A with A⊕1, open the arms, corresponding to the distributive law (div. power 3), and then establish power 4 and 5.

butt

.

Determine the Zhegalkin polynomial of the function f(X,Y)=X→Y

Decision

1. (method of insignificant coefficients).
Let's write down the search polynomial in the form:




P=c 0 ⊕c 1 x⊕c 2 y⊕c 12 xy