Find out the graphical superposition of lines. Important functions. Monotonic Boolean functions

Topic: “Function: concepts, methods of implementation, main characteristics. Gate function. Superposition of functions.

Lesson epigraph:

“Wing now and don’t worry about

vivchenim – absolutely shady.

Meek over the chimos without screwing it up

in advance the subject of thought -

Confucius.

Meta and psychological and pedagogical instructions for the lesson:

1) Behind-the-scenes lighting (normative) meta: Repeat with students the importance and power of functions. Introduce the concept of superposition of functions.

2) Department of Mathematical Development for Students: on non-standard elementary-mathematical material, continue the development of the mental understanding of students, the substitute cognitive structure of their mathematical intelligence, including the range of logical-deductive and inductive, analytical and synthetic ethical reverse thinking, to algebraic specification, to reflection and independence as metacognitive ability of students ; to continue the development of a culture of written and oral communication as psychological mechanisms for elementary-mathematical intelligence.

3) Vikhovny Zavodnya: continue to especially instill in students a cognitive interest in mathematics, competence, sense of commitment, academic independence, communicative skills, work with a group, a student, and peers; autogogic creation to the level of elementary-mathematical activity, aiming for high and significant results (acmeic motive).

Lesson type: introduction of new material; for the criterion of a solid mathematical course - a practical lesson; according to the criterion of the type of informational interaction of educational and output - a lesson in education.

Lesson instruction:

1. Basic literature:

1) Kudryavtsev mathematical analysis: Head. for students of universities and universities. U 3 t. T. 3. - 2nd version, Revised. ta add. - M.: Vishch. school, 1989. - 352 p. : ill.

2) Demidovich is responsible for mathematical analysis. - 9th type. - M.: Vidavnitstvo "Science", 1977.

2. Illustrations.

Lesson progress.

1. Stunned by those and the main illumination of the lesson; stimulation of a sense of commitment, relevance, and interest to students during preparation before the session.

2.Repetition of the material with food.

a) Dates of assignment of the function.

One of the main mathematical understandings is the concept of functions. The concept of function is related to the established position between the elements of two multipliers.

Let's give two unempty factors and . Type f, as the skin element consists of one and only one element is called function it is written y = f(x). In other words, the function f depicts impersonal on impersonal.

https://pandia.ru/text/79/018/images/image003_18.gif" width="63" height="27">.gif" width="59" height="26"> called meaningless the function f is denoted by E(f).

b) Numerical functions. Function graph. Methods for setting functions.

Let the function be given.

If the elements of multiplicities are decimal numbers, then the function f is called numerical function . Zminna x whereby it is called argument or independent exchangeable, and y – function or else stale meat(View x). It seems that the values ​​x and y themselves are in functional position.

Function graph y = f(x) is called without all points of the Oxy plane, each of which x is the value of the argument, and y is the corresponding value of the function.

To set up the function y = f(x), you need to specify a rule that allows, knowing x, to find similar values ​​for y.

Most often there are three ways to perform a function: analytical, tabular, graphic.

Analytical method: the function is specified in the form of one or several formulas or equations.

For example:

Since the area of ​​significance of the function y = f(x) is not assigned, it is transferred so that it is avoided without any meaning to the argument, for which the corresponding formula has a sense.

The analytical method of defining a function is the most thorough, since previously applied methods of mathematical analysis allow us to completely trace the function y = f(x).

Graphic method: The function schedule is set.

The advantage of a graphic design is its precision, not so much its imprecision.

Tabular method: the function is designated by a table with a row of argument values ​​and subordinate function values. For example, the tables contain the values ​​of trigonometric functions and logarithmic tables.

c) Main indicators of function.

1. The function y = f(x), calculated on the multiplier D, is called steam rooms how to think about it f(-x) = f(x); unpaired How to think about it f(-x) = -f(x).

The graph of a paired function is symmetrical along the Oy axis, and that of an unpaired function is symmetrical along the coordinates. For example, – male functions; and y = sinx, - functions of the legal form, then not guys and not guys.

2. Let the function y = f(x) be calculated by the multiplier D and let it go. Whatever the meaning of the arguments from inequality, inequality emerges: , then the function is called growing on impersonality; yakscho , then the function is called non-falling on https://pandia.ru/text/79/018/images/image021_1.gif" sound function. subsiding on the; - immature .

Growing, non-growing, decreasing and immutable functions on the multiplier D are the values ​​(x+T)D and equalizes f(x+T) = f(x).

To generate a graph of a periodic function, it is enough for the period T to wake it up for any segment until T and periodically extend it over the entire designated area.

The basic power of the periodic function is significant.

1) The algebraic sum of periodic functions that cover the same period T is a periodic function with period T.

2) Since the function f(x) is the period T, then the function f(ax) is the period T/a.

d) The function is wrapped.

Let a function y = f(x) be given with a valued region D and an immutable value E. This function z(y) is called gateway to the function f(x) and is written in the following form: . About the functions y = f(x) and x = z(y) we can say that they are mutually inverse. To know the function x = z(y) wrapped into the function y = f(x), it is enough to calculate the equation f(x) = y to x.

Apply it:

1. For the function y = 2x, the reverse function is the function x = y;

2. For function The return function is a function.

The vicious vigor of the functions of the vibiva, the functions y = f (x) may be visible, if it is a mentally unambiguous man, the many in the many d I e. Zvidsey, be a bloom a strictly monotonic function has a reversal . In this case, as the function grows (changes), then the return function also grows (changes).

3. Introduction of new material.

Folding function.

Let the function y = f(u) be assigned to the multiplier D, and the function u = z(x) be assigned to the multiplier , and at the same time . Then the multiplier is defined by the function u = f(z(x)), which is called folding function view x (or superposition function assignments, or function as function ).

The value u = z(x) is called intermediate argument folding functions.

For example, the function y = sin2x is a superposition of two functions y = sinu and u = 2x. A folding function can take a number of intermediate arguments.

4. Version of several butts for the board.

5. Recap of the lesson.

1) theoretical and applied bags of practical employment; differentiated assessment of the level of mental assessment of students; the level of competence acquired by them, the quality of oral and written mathematical language; the level of revealed creativity; the level of independence and reflection; level of initiative, knowledgeable interest to other methods of mathematical thinking; competitiveness, intellectual excellence, dedication to tall displays primary mathematical activities;

2) stupefaction of argumentative notes, lesson ball.

Use the function

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Advantages of daily schedules online

• Visual representation of the functions to be introduced
• Pobudova even folding graphs
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• The ability to save graphics and post messages on them, making them available to everyone on the Internet.
• Controlling scale and line color
• Possibility of weekly graphs behind points, vicor constants
• Call up a number of graphics functions at the same time
• Pobudova graphs in the polar coordinate system (Vikorist r and θ(\theta))

We can easily provide you with graphics of varying complexity online. Pobudova get lost in mittevo. Request service for finding points of the cross function, displaying graphs for their further movement to Word document as illustrations of the current task, to analyze the behavioral features of function graphs. The optimal browser for working with graphs on this page Google Chrome. In case of other browsers, the correctness of the robot is not guaranteed.

The function f, supported by the function f 1 , f 2 ... f n with the additional operation of substitution and renaming of arguments, is called superposition functions.

Any formula that expresses a function as a superposition of other functions specifies a method for calculating it, so that the formula can be calculated by calculating the values ​​of its subformulas. The values ​​of the formula can be calculated using a given set of double values.

According to the skin formula, you can update the table of logical functions, but by mistake, because skin logical functions can be detected in a number of formulas in different bases

Formulas F i and F j representing one and the same logical function f i are called equivalent . So, with equivalent formulas:

1. f 2 (x 1 ; x 2)=(x 1 ×`x 2)=ù(`x 1 Úx 2)= ù(x 1 ®x 2);

2. f 6 (x 1 ; x 2)=(`x 1 ×x 2 Úx 1 ×`x 2)= u(x 1 “x 2)=(x 1 Åx 2);

3. f 8 (x 1 ; x 2) = (x 1 × x 2) = u (x 1 Úx 2) = (x 1 x 2);

4. f 14 (x 1 ;x 2)=(`x 1 Ú`x 2)= ù(x 1 ×x 2)=x 1 ½x 2 ;

5. f 9 (x 1 ;x 2)=((`x 1 ×` x 2)Ú(x 1 ×x 2))=(x 1 “x 2);

6. f 13 (x 1 ;x 2)= (`x 1 Úx 2)=(x 1 ®x 2).

If a formula F is a subformula F i , then replacing F i with an equivalent F j does not change the value of the formula F for any set of Boolean vectors, but rather changes the form of its description. The formula F' is again determined to be equivalent to the formula F.

To simplify the complex expressions of the algebra of the Boolean function, concatenate equivalent transformations , Vikorist laws of Boolean algebra substitution rules і substitution ,

When writing Boolean algebra formulas, remember:

· The number of left arms is equal to the number of right arms,

· There are no two logical connections to stand on, so between them the formula is to blame,

· No two orders vartich formulas, then there is a logical connection between them,

· the logical connection “×” is stronger than the logical connection “Ú”,

· if “ù” is added to the formula (F 1 ×F 2) or (F 1 Ú F 2), then first of all the viconation of de Morgan’s law: ù(F 1 ×F 2) = `F 1 Ú ` F 2 or ù(F 1 ÚF 2)=`F 1 ×`F 2 ;

· Operation “ × ” is stronger than the “Ú”, which allows you to lower the arms.

butt: viconati equivalent rearrangement of the formula F = x 1 x x 2 x x 3 x x 4 Ú x 1 x x 3 Ú x 2 x x 3 Ú x 3 x 4 .

· Behind the law of commutativity:

F = x 3 × x 1 × x 2 × 4 Úx 3 × 1 Úx 3 × 2 Úx 3 × 4;

· Behind the law of distributivity:

F=x 3 ×x 1 ×x 2 ×`x 4 Úx 3 ×`x 1 Úx 3 ×(`x 2 Úx 4);

· Behind the law of distributivity:

F=x 3 ×x 1 ×x 2 ×`x 4 Úx 3 ×(`x 1 Ú`x 2 Úx 4);

· Behind the law of distributivity:

F=x 3 ×((x 1 ×x 2 ×` x 4)Ú(`x 1 Ú`x 2 Úx 4));

· Behind De Morgan's law:

F=x 3 ×((x 1 ×x 2 ×`x 4)Úù(x 1 ×x 2 ×`x 4));

· Behind the law of protirichchya:

Thus x 1 x 2 x 3 x 4 x 4 x 1 x x 3 Ú x 2 x x 3 Úx 3 x x 4 = x 3 .

Butt: Viconati reformulation of the formula

F=(x 1 ×`x 2 Ú`x 1 ×x 2)×ù(x 1 ×x 2)Ú(x 1 ×x 2)×ù(x 1 ×`x 2 Ú`x 1 ×x 2 );

· Behind De Morgan's law

F=(x 1 ×`x 2 Ú`x 1 ×x 2)×(`x 1 Ú`x 2)Ú(x 1 ×x 2)×(`x 1 Úx 2)×(x 1 Ú`x 2);

· Behind the law of distributivity:

F=x 1 ×`x 2 Ú`x 1 ×x 2 Úx 1 ×x 2;

· Behind the laws of commutativity and distributivity:

F= `x 1 ×x 2 Úx 1 ×(`x 2 Úx 2);

· Behind the law of protirichchya:

F = x 1 × x 2 Úx 1;

· Behind Poretsky's law

In this order (x 1 ×`x 2 Ú`x 1 ×x 2)×ù(x 1 ×x 2)Ú(x 1 ×x 2)×ù(x 1 ×`x 2 Ú`x 1 ×x 2 )= (x 2 Úx 1).

Butt: Viconati reformulation of the formula F=ù(`x 1 Úx 2)Ú((`x 1 Úx 3)×x 2).

· Behind De Morgan's law:

F= ù(`x 1 Úx 2)×ù((`x 1 Úx 3)×x 2);

· Behind De Morgan's law:

F=x 1 ×`x 2 ×(ù(`x 1 Úx 3)Ú`x 2);

· Behind De Morgan's law:

F=x 1 ×`x 2 ×(x 1 ×`x 3 Ú`x 2);

· Behind the law of distributivity:

F=x 1 ×`x 2 ×`x 3 Úx 1 ×`x 2;

· Follow the law:

In this manner?

butt: Viconati re-created formula:

F=ù(x 1 ®x 2)×(`x 3 U`x 4)Ú(x 1 x 2)×ù(x 3 x 4).

1) convert the formula to the basis of Boolean algebra:

F=ù(`x 1 Úx 2)×(`x 3 Ú`x 4)Úù(x 1 Úx 2)× ù(x 3 ×x 4);

2) lower the “`” sign to the doubles:

F=(x 1 ×`x 2)×(`x 3 Ú`x 4)Ú(`x 1 ×`x 2)×(`x 3 Ú`x 4);

3) transform the formula using the law of distributivity:

F = x 1 × x 2 × x 3 Úx 1 × x 2 × x 4 Ú x 1 × x 2 × x 3 Ú x 1 × x 2 × x 4;

4) blame the bow `x2 for the law of distributivity:

F=`x 2 ×(x 1 ×`x 3 Úx 1 ×`x 4 Ú`x 1 ×`x 3 Ú`x 1 x`x 4);

5) transform it according to the law of distributivity:

F=`x 2 ×(`x 3 ×(x 1 Ú`x 1)Ú`x 4 ×(x 1 Ú`x 1));

6) vikoristovyvat the law of protirіchchya:

F=`x 2 ×(`x 3 Ú`x 4)

The power of Boolean functions

Nutrition often comes up: could a Boolean function be represented by a superposition of the formulas f 0 , f 1, .. f 15? In order to evaluate the possibility of forming any Boolean function behind another superposition of these formulas, it is necessary to consider their power and intelligence of a functionally new system.

Self-propelled Boolean functions

self-propelled , if f(x 1 ;x 2 ;…x n)=`f(`x 1 ;`x 2 ;…` x n).

For example, functions f 3 (x 1 ;x 2)=x 1 , f 5 (x 1 ;x 2)=x 2 , f 10 (x 1 ;x 2)=`x 2 and f 12 (x 1 ;x 2)=`x 1 are self-contained, because when you change the value of the argument, they change their value.

Any function separated by the additional operation of superposition from self-double Boolean functions is itself self-double. Therefore, the absence of self-contained Boolean functions does not allow the formation of non-self-contained functions.

Monotonic Boolean functions

The function f(x 1; x 2; … x n) is called monotonous , because for the skin s 1i £s 2i Boolean vectors (s 11 ; s 12 ;……; s 1n) i (s 21 ;s 22 ;……; ;s 1i ;…;s 1n)£f(s 21 ; s 22 ;…;s 2i ;…;s 2n).

For example, for the function f(x 1 ; x 2) monotonic functions e:

if (0; 0) £ (0; 1), then f(0; 0) £ f (0; 1),

if (0; 0) £ (1; 0), then f(0; 0) £ f(1; 0),

if (0; 1) £ (1; 1), then f(0; 1) £ f(1; 1),

if (1; 0) £ (1; 1), then f(1; 0) £ f(1; 1).

Such minds are satisfied with the following functions:

f 0 (x 1; x 2) = 0; f 1 (x 1 ; x 2) = (x 1 × x 2); f 3 (x 1; x 2) = x 1; f 5 (x 1; x 2) = x 2; f 7 (x 1; x 2) = (x 1 Úx 2); f 15 (x 1; x 2) = 1.

Whether a function is separated by an additional operation of superposition of monotonic Boolean functions, itself is monotonic. Therefore, the absence of monotonous functions does not allow the formation of non-monotonic functions.

Linear Boolean functions

The Zhegalkin algebra, which expands onto the basis F 4 = (×; Å; 1), allows any logical function to be represented by a polynomial, the member of which is the conjunction of I Boolean variables of a Boolean vector between 0£i£n:

P(x 1 ; x 2 ;…x n)=b 0 ×1 Å b i ×x i Å 1 j j k £ n b j ×x j ×x k Å…… 2n-1 ×x 1 ×x 2 ×... ×x n.

For example, for logical functions f 8 (x 1 ; x 2)

The Zhegalkin polynomial looks like this: P(x 1 ; x 2)=1Å x 1 Å x 2 Å x 1 ×x 2 .

The advantages of Zhegalkin algebra lie in the “arithmetization” of logical formulas, and the shortcomings lie in complexity, especially for the large number of double changes.

Zhegalkin's polynomials, which replace the conjunction of two-dimensional variables, then. P(x 1 ; x 2 ;…;x n)=b 0 ×1Åb 1 ×x 1 Å…Åb n ×x n name linear .

For example, f 9 (x 1 ; x 2) = 1Åx 1 Åx 2 or f 12 (x 1 ; x 2) = 1Åx 1 .

The main power of the operation added to module 2 is indicated in table 1.18.

Since the logical function is given by the table and formula for the skin basis, then. Given the values ​​of a Boolean function for different sets of Boolean variables, then you can calculate all

coefficients b i of the Zhegalkin polynomial, combining the system of ranks for all known sets of double variables.

butt: Given a Boolean function f(x 1 ;x 2)=x 1 Úx 2 . The values ​​of this function are visible across all sets of Boolean variables.

F(0;0)=0=b 0 ×1Å b 1 ×0 Å b 2 ×0 Å b 3 ×0×0;

f(1;0)=1=b 0 ×1Å b 1 ×1Å b 2 ×0Å b 3 ×1×0;

f(1;1)=1=b 0 ×1Å b 1 ×1Å b 2 ×1Å b 3 ×1×1;

Signs are known b 0 = 0; b 1 = 1; b 2 = 1; b 3 =1.

Also, (x 1 Úx 2) = x 1 x 2 x 1 x 2, then the disjunction is a nonlinear Boolean function.

butt: given a Boolean function f(x 1 ;x 2)=(x 1 ®x 2). The meanings of these functions are also the same for all sets of double changers.

F(0;0)=1=b 0 ×1Å b 1 ×0 Å b 2 ×0 Å b 3 ×0×0;

f(0;1)=1=b 0 ×1Å b 1 ×0 Å b 2 ×1Å b 3 ×0×1;

f(1;0)=0=b 0 ×1Åb 1 ×1Åb 2 ×0Åb 3 ×1×0;

f(1;1)=1=b 0 ×1Åb 1 ×1Åb 2 ×1Åb 3 ×1×1;

Stars are known b 0 = 1; b 1 = 1; b 2 = 0; b 3 =1.

Otzhe, (x 1 ®x 2) = 1Å x 2 Å x 1 ×x 2.

Table 1.19 shows the Zhegalkin polynomials for the main representatives of the Boolean functions from Table 1.15.

Once an analytical expression of a logical function and its unknown value is given for different sets of double variables, then one can construct a Zhegalkin polynomial, spiraling onto the conjunctive algebra basis Boolean F 2 =(` ; ×):

Let f(x 1 ; x 2)=(x 1 Úx 2).

Todi (x 1 Úx 2)=ù(`x 1 ×`x 2)=((x 1 Å 1)×(x 2 Å 1))Å 1=x 1 ×x 2 Å x 1 ×1Å x 2 × 1Å 1×1Å1=

(x 1 x 2 x 1 x 2).

Let f(x 1 ;x 2)=(x 1 ®x 2).

Todi (x 1 ®x 2)=(`x 1 Úx 2)=ù(x 1 ×`x 2)=x 1 ×(x 2 Å 1)Å 1=x 1 ×x 2 Å x 1 ×1Å 1 = =(1Åx 1 Åx 1 ×x 2).

Let f(x 1; x 2) = (x 1 “x 2).

Todi (x 1 “x 2)=(`x 1 ×`x 2 Úx 1 ×x 2)=ù(ù(`x 1 ×`x 2)×ù(x 1 ×x 2))=((( ( x 1 Å1)×(x 2 Å1))Å1)× ×(x 1 ×x 2 Å)Å1=(x 1 ×x 2 Åx 1 Åx 2 Å1Å1)×(x 1 ×x 2 Å1)Å1=x 1 ×x 2 Åx 1 ×x 2 Åx 1 ×x 2 Åx 1 Å

x 1 ×x 2 Åx 2 Å1=(1Åx 1 Åx 2).

Any function that is separated by an additional operation of superposition from linear logical functions is itself linear. Therefore, the absence of linear functions does not allow the formation of nonlinear functions.

1.5.6.4. Functions that are saved “0”

The function f(x 1 ; x 2 ;...x n) is called save “0”, when the value of the double change (0; 0;...0) is set, the function receives the value f(0; 0;…0)=0.

For example, f 0 (0; 0) = 0, f 3 (0; 0) = 0, f 7 (0; 0) = 0 and in.

Any function that is removed by an additional superposition operation with a function that stores “0” is itself a function that stores “0.” Therefore, no function that stores “0” is not allowed to be formatted functions that do not store "0".

1.5.6.5. Functions that are saved “1”

The function f(x 1 ; x 2 ;…x n) is called save “1”, since by typing the values ​​of double change (1; 1;…1) the function takes the value f(1;1;…1)=1.

For example, f 1 (1; 1) = 1, f3 (1; 1) = 1, f 5 (1; 1) = 1 and in.

Any function that is separated by a superposition operation with a function that stores "1" itself stores "1". Any functions that store "1" are not allowed to formulate functions that do not store "1".

Single-ended (which does not replace memory elements) discrete logic devices implement at the output a certain set of logic algebra functions `F m =(F 1 ,F 2 ,…,F m), which at any given time should only lie outside the entrance to the building x n =(x 1 ,x 2 ,…, x n): `F m = `F m(`x n). In practice, such devices are designed and manufactured from a number of indivisible elements in order to implement a single dialing (system) ( f) elementary functions of algebra by connecting the outputs of some elements to the inputs of others.

When designing a bath logical devices What is relevant is nutrition.

1. A system of elementary functions is specified ( f). What are the output functions? F i you can remove the vikory functions from ( f}?

2. No output Boolean functions are specified ( F) (zokrem, equal to all impersonal functions of the logic of algebra R 2). What is the output system of elementary functions ( f), which ensures the possibility of extracting the output from the multiplying function ( F}?

For the connected circuit to this power supply, the concepts of superposition, closedness and recurrence of systems of functions are used.

Viznachennya. Let's take a look at the meaningless logical connections ( F), which indicates the singing system function ( f} . Superposition over{f) is called a function j, which can be implemented by a formula over ( F}.

Another superposition is possible as a result of substituting a function with ( f) as the arguments of the function are completely impersonal.

Butt 1. Let's take a look at the system of functions ( f} = {f 1 (X) =`x, f 2 (x,y)= X&y, f 3 (x,y)=XÚ y). Substituting a function f 3 (x,y) instead of the first argument X function f 1 (X), replacing another - f 2 (x,y), we cancel the superposition h(x,y)=f 3 (f 1 (X),f 2 (x,y))=`xÚ X& at. The physical implementation of the substitution is given in Fig. 1.18.

Viznachennya. Let's go M-decade of impersonal functions of the algebra of logic ( P 2). The impersonality of all superpositions over M called muttered impersonality M and is indicated by [ M]. Otrimannya [ M]behind the output factor M called closing operation. Bezlich M called functionally closed class, yakscho [ M] = M. Submultiple mÍ M called functionally new system in M, yakscho [ m] = M.

Zamikannya [ M]is all the impersonal functions that can be eliminated from M by the way of the operation of superposition, then. all possible substitutions.

Respect. 1. Obviously, be it a system of functions ( f) is functionally complete in itself.

2 . Without compromising strength, it is important to understand what the same function is f(X)=x, which does not change the value of the truth of the changes, initially enters the warehouse of any system of functions.

Butt 2. For the systems of functions discussed below ( f) Vikonati such actions:

1) know the sound [ f],

2) z'yasuvati, chi will be the system ( f) closed class,

3) know the functionalities of all systems in ( f}.

Decision.

I. ( f}={0} . At the hour when the function is installed ( fº 0) I’ll take it away from myself, then. No new functions are being created. The star screams: [ f] = {f). The system is viewed as a functionally closed class. The system in it is functionally new and modern throughout ( f}.

ІІ. ( f} = {0,Ø } . Substitution Ø (Ø X) gives the same function, but does not formally expand the output system. However, when substituting Ø (0), we subtract the same unit - new function, Which the output system did not have: Ø (0)=1 . The suspension of other settings should not lead to the appearance of new functions, for example: ØØ 0 = 0, 0(Ø X)=0.

In this way, the establishment of the superposition operation made it possible to remove more equal parts from the external impersonal function [ f]=(0,Ø ,1). Zvidsi viplyaet suvore entry: ( f} Ì [ f]. Vikhіdna system ( f) is not a functionally closed class. Cream of the system itself ( f)other functional new systems It doesn’t have any, but there are several sounds with one function. f= 0 cannot be subtracted by substitution, and the same zero cannot be subtracted from the same function.

ІІІ. ( f) = (& ,Ú ,Ø ).The closures of this system are all functions of algebraic logic P 2, since the formula of either of them can be represented in terms of DNF or CNF, which have elementary functions ( f) = (& ,Ú ,Ø). This fact is a constructive proof of the completeness of the considered system of functions P 2: [f]=P 2 .

Oskolki in P 2 take place without other functions, subordinated to ( f) = (& ,Ú ,Ø ), then the result is the following: ( f}Ì[ f]. The system is no longer a functionally closed class.

In addition to the system itself, functionally it will have subsystems ( f) 1 = (& ,Ø ) that ( f) 2 = (Ú, Ø). Therefore, using De Morgan's rules, the logical addition function can be expressed through (& ,Ø), and the logical multiplication function & - through (Ú, Ø):

(X & at) = Ø (` XÚ` at), (X Ú at) = Ø ( X &`at).

Other functionally advanced subsystems ( f) No.

Checking the completeness of the function subsystem ( f) 1 М ( f) for the entire system ( f) can be selected by the way of viewing ( f) 1 before the other, obviously again in ( f) system.

Inconsistency of the subsystem ( f) 1 in ( f) can be verified by completing the entry [ f 1 ] М [ f].

Viznachennya. Submultiple mÍ M call functional basis(basis)M system, yakscho [ m] = M, and after turning off any function from it, it’s impossible to solve again M .

Respect. Bases of a system of functions (f) all of them are functionally advanced subsystems (f) 1, which is impossible to change without wasting money (f).

Butt 3. For all systems examined in Appendix 2, you can know the basis.

Decision.In types 1 and 2, the functionality is different except for the systems themselves and their sound is impossible. Well, it smells like bases.

In case 3 there are two functionally new ones ( f) subsystems ( f) 1 = (&,Ø) and ( f) 2 =(Ú,Ø), which is impossible to speed up without wasting time. The stench will be the basis of the system ( f} = {&,Ú,Ø}.

Viznachennya. Let the system go ( f) is a closed class. This subdivision ( f) 1 М ( f)name first class{f), yakshcho ( f) 1 not exactly in ( f} ([f 1 ] М [ f]), and for any function of the system( f), do not enter until ( f) 1 (jО( f} \ {f) 1) true: [ jÈ { f} 1 ] = [f], then. supplement jk ( f) 1 to work again in ( f} .

Zavdannya

1. Check the closedness of multipliers with functions:

a) (Ø); b) (1, Ø); c) ((0111); (10)); d) ((11101110); (0110)); d) ((0001); (00000001);

2. Check the completeness of the systems functions P 2:

a) (0,Ø); b) ((0101), (1010)); V) (?); d) ((0001), (1010)).

3. Find out the closure of the system of functions and its basis:

a) (0, 1, Ø); b) ((1000), (1010), (0101)); c) ((0001), (1110), (10)); d) ((1010), (0001), (0111)).

1.10.2 Functions that save constants. Klasi T 0 and T 1

Viznachennya. Function f(`x n) saves 0, yakscho f(0,..., 0) = 0. Function f(`x n) saves 1, yakscho f(1, ... , 1) = 1.

Impersonal function n The variables that save 0 and 1 mean, obviously, T 0 nі T 1 n. All multiplicities of logic algebra functions that save 0 and 1 , signify T 0 і T 1 . Kozhna z mnozhin T 0 that T 1 є closed front class R 2 .

With elementary functions T 0 that T 1 enter at the same time, for example, і Ú. Belonging of any function to classes T 0 , T 1, you can check the first and remaining values ​​of the vector value in the truth table or by substituting zeros and one in the formula with an analytical specified function.

Viznachennya.Double This is called a substitution if, instead of many independent variables, you substitute that same variable in a function. Given the magnitude of the changes in the sets, which previously acquired values ​​independently of each other, will now be the same.

ZAVDANNYA

1.Check ownership to classes T 0 і T 1 functions:

a) regular folding, b) regular multiplying, c) constants, d) xyÚ yz d) X® at® xy, e) XÅ at, and) ( X 1 Å … Å X n) ® ( y 1 Å … Å y m) at n,mÎ N.

2. Bring closeness of the skin from classes T 0 і T 1 .

3. Bring what you want f(`x n) Ï T 0 then with it, using the path of a duplicate substitution, you can subtract the constant 1 or the interchange.

4. Bring what you mean f(`x n) Ï T 1 then with it, using the path of a duplicate substitution, you can subtract the constant 0 or the enumeration.

5. Improve skin tone T 0 і T 1 (for example, upgrading the upgraded system to ( f} = {& ,Ú ,Ø }).

6. Know the strength of classes T 0 nі T 1 n.

We are familiar with the concept of superposition (or overlay) of a function, which means that instead of an argument, a function is substituted with a function in another argument. For example, the superposition of a function gives the function the same output as the function

In the conventional view, it is acceptable that the function is designated in the given area and the function is designated in the area and the meanings are all located in the area of ​​the same function

When given a starting point, find the corresponding value (as a rule, which is characterized by the sign of the value of y), and then set the corresponding value of y (usually,

the meaning that is characterized by the sign, i is respected as similar to the symbol x. A function is separated from a function or a complex function is the result of a superposition of a function

It is assumed that the important function does not go beyond those areas in which the function is already assigned: if you omit it, then you can go out of focus. For example, those with respect can see only such meanings, which would otherwise be of little sense.

It is important for us to point out that the characteristic of a function as foldable is not related to the nature of the functional location of the type, but only to the method of assigning this location. For example, let's do it for

Here the function appeared to be specified as a folding function.

Now, if the concept of superposition of a function is clearly understood, we can accurately characterize the simplest of these classes of functions that are included in the analysis: first of all, a list of more elementary functions, and then They all have to do with the help of four arithmetic action and superposition, sequentially stasted the final number of times. To say about them, the stench manifests itself through elementary sight; Sometimes they are also called elementary.

Nowadays, based on a complex analytical apparatus (infinite series, integrals), we know other functions that also play significant roles in analysis, but also fall within the class of elementary functions.