Independent work on logic. Test on computer science and ICT "elements of algebra of logic" Let Anna like the lessons

Ivanov, Ozhegov, Krysin, Lopatin, Bunin, Fonvizin, Green, Tseitlin, Darwin. Male surnames -ov, -in(Russian and borrowed) are inclined: Ozhegov's dictionary, the story belongs to Bunin, I'm waiting for Ivanov, talking with Krysin about Green.

Borrowed surnames -ov, -in which belong foreigners, in the form of the instrumental case they have an ending -ohm(as nouns of the second school declension, for example table, table): the theory was proposed by Darwin, the film was directed by Chaplin, the book was written by Cronin.(Interestingly, the pseudonym is also inclined Green, belonging to a Russian writer: the book has been written Green.) Homonymous Russian surnames have the ending - th in the instrumental case: with Chaplin(from the dialect word Chaplya"heron"), with Kronin(from crown).

Shukshina, Ilyina, Petrova, Fedorova, Graudina. Women's surnames -ina, -ova bow down. Surnames like Currant, Pearl Declined in two ways, depending on the declination of the male surname (Irina Zhemchuzhina and Irina Zhemchuzhina, Zoya Smorodina and Zoya Smorodina). If the man's surname is Zhemchuzhin, then correct: arrival of Irina Zhemchuzhina. If the man's surname is Pearl, then correct: arrival of Irina Zhemchuzhina(surname is declined as a common noun pearl).

Okudzhava, Globa, Shcherba, Vayda -a, -i unstressed, usually inflected (songs by Bulat Okudzhava, forecasts by Pavel Globa, films by Andrzej Wajda).

Gamsakhurdia, Beria, Danelia, Pihoya. Surnames starting with - ia do not bow: books by Konstantin Gamsakhurdia. In contrast, Georgian surnames start with - and I inflected: Beria's crimes, Danelia's films. Surnames starting with - oh inflected after the noun needles: about Rudolf Pihoy.

Mitta, Frying pan, Kvasha. Male and female surnames of Slavic origin in -A percussion bow (about Alexander Mitte, with Grigory Skovoroda, with Igor Kvasha).

Dumas, Zola, Gavalda. Male and female surnames of French origin in -a, -i drums don't bow (books by Alexandre Dumas, Emile Zola, Anna Gavalda).

Korolenko, Dovzhenko, Shevchenko, Lukashenko, Petrenko. Male and female surnames -ko don't bow down.

Falcone, Gastello, Zhivago, Durnovo, Lando, Dante, Koni. Male and female surnames -o, -e, -i, -u, -yu don't bow down.

Blok, Gaft, Brockhaus, Hasek, Mickiewicz, Stirlitz, Dahl, Rosenthal, Gudziy, Macbeth. Male surnames ending in a consonant, whether soft or hard, are declined (Brockhaus encyclopedia, poetry by Mickiewicz, dictionary written by Dahl). Female surnames ending in a consonant are not declined (to Lyubov Dmitrievna Blok, memoirs of Nadezhda Mandelstam).

Black, Red, Twisted. Male and female surnames -y, -them don't bow down.

Kalakutskaya, Kalakutsky, White, White, Tolstaya, Tolstoy. Male and female surnames, which are adjectives in form, are declined according to the adjective model:

I. p.: Andrey Bely, Tatyana Tolstaya

R. p.: Andrey Bely, Tatyana Tolstoy

D. p.: Andrey Bely, Tatyana Tolstoy

V. p.: Andrey Bely, Tatyana Tolstoy

T.p.: with Andrei Bely, with Tatyana Tolstaya

P. p.: about Andrei Bely, about Tatyana Tolstoy

Russia is a multinational country, so there are many names and surnames of different origins.

We have to sign notebooks, fill out documents, and we must put our last name in a certain case and not make a mistake with the ending. This is where difficulties await us. For example, how to say correctly: “reward Lyanka Elena or Lyanka Elena, Bavtruk Timur or Bavtruk Timur, Anton Sedykh or Anton Sedogo»?

Today we will try to understand some aspects of the declension of foreign and Russian-speaking surnames, male and female.

Let's start with the fact that most of the surnames are originally Russian similar in form to adjectives with suffixes -sk-, -in-, -ov- (-ev-): Hvorostovsky, Veselkin, Mikhalkov, Ivanov, Tsarev. They can take the form of both male and feminine, and also used in the plural. At the same time, rarely does anyone have difficulties with declension of such surnames.

I. p. (who? what?) Hvorostovsky, Hvorostovskaya, Hvorostovsky.

R. p. (who? what?) Hvorostovsky, Hvorostovskaya, Hvorostovsky.

D. p. (to whom? what?) Hvorostovsky, Hvorostovskaya, Hvorostovsky.

V. p. (who? what?) Hvorostovsky, Hvorostovskaya, Hvorostovsky.

etc. (by whom? with what?) Hvorostovsky, Hvorostovskaya, Hvorostovsky.

P. p. (about whom? about what?) about Hvorostovsky, about Hvorostovskaya, about the Hvorostovskys.

However, you need to be careful with surnames ending with a consonant or a soft sign. For example, Jackal, Tavgen, Korob, Great-grandfather. In this case, the declination will depend on What gender does the surname belong to? If we're talking about about a woman, they are similar e surnames are indeclinable, but male surnames are indeclinable, like nouns of the 2nd declension husband. r. (such as table, deer). This does not apply to last names ending in - them(s). For example, go along with Jackal Anna And Jackal Anton, talk about Tavgen Anastasia and about Tavgena Alexandra, walk with Daria's great-grandfather and with Great-grandfather Emelyan.

Some surnames like Child, Kravets, Zhuravel may have variable declension due to the fact that they are similar to common nouns. When declension of nouns occurs dropping a vowel at the end of a word(zhur flight ow I, bathe the rebbe NK a), when declining a surname, the vowel can be preserved to prevent distortion or comical sound of the surname (write Zhuravel, dispatch from child).

Don't bow male and female surnames -s(s). talk about Diana Sedykh and about Anton Sedykh, write Velimiru Kruchenykh And Antonina Kruchenykh.

All female and male surnames ending in vowels, except -A or -I, are unyielding. For example, Artman, Amadou, Bossuet, Goethe, Galsworthy, Gramsci, Grétry, Debussy, Dzhusoit, Daudet, Camus, Cornu, Lully, Manzu, Modigliani, Navoi, Rustaveli, Ordzhonikidze, Chabukiani, Enescu and many others.

This also includes surnames ending in -O, and last names Ukrainian origin on -ko. For example, Hugo, La Rochefoucauld, Leoncavallo, Longfellow, Picasso, Craft, Khitrovo, Chamisso, Makarenko, Korolenko, Gorbatko, Shepitko, Savchenko, Zhivago, Derevyago, etc.

Declension of surnames ending in -A, causes the greatest difficulties. Here it is necessary to take into account several criteria: origin of the surname, stress and letter after which -A located. Let's try to simplify the picture as much as possible.

Surnames do not lean towards -A, if this letter is preceded by a vowel (most often at or And): Gulia, Moravia, Delacroix, Heredia. This also applies to last names of Georgian origin.

Surnames do not lean towards -AFrench origin with emphasis on the last syllable: Degas, Dumas, Luc, Thomas, Fermat, Petipa etc.

All other surnames are -A declined in Russian. Bring Lyanka Elena, take from Shatravki Inna, read Petrarch, along with Kurosawa, O Glinka, For Alexandra Mitta.

The situation is similar with the declination of surnames from the final -I: surnames are not declined French origin with emphasis on the last syllable (Zola). All other surnames ending in -I, bow. For example, persuade Ivan Golovnya And Elena Golovnya, write about Beria, movie Georgy Danelia.

Thus, as you may have noticed, you need to know not so many rules in order to correctly pronounce your surname in Russian. We hope that now you will not make mistakes when signing a notebook or filling out documents! But if you still have any doubts, please contact us. Our specialists will always try to help!

Good luck to you and the beautiful, literate, rich Russian language!

website, when copying material in full or in part, a link to the original source is required.

1. Names (Slavic) on -O such as Levko, Marko, Pavlo, Petro are declined according to the model of the declension of masculine-neuter nouns, for example: in front of Levka, in Mark; M. Gorky does not decline the name Danko (“... she spoke about Danko’s burning heart”).

Names having parallel forms on -O – -A(Gavrilo - Gavrila, Mikhaila - Mikhaila), usually declined according to the type of nouns of the feminine declension: at Gavrila, to Gavrila, with Gavrila. Other endings (at Gavril, to Gavril, with Gavril) are formed from another initial form Gavril.

2. Foreign names the consonant sound is inclined regardless of whether they are used independently or together with the surname, for example: the novels of Jules Verne (not “Jules Verne”), the stories of Mark Twain, the plays of John Boynton Priestley, the fairy tales of Hans Christian Andersen, the book of Pierre-Henri Simon. Partial deviations are observed with double French names, for example: the philosophical views of Jean-Jacques Rousseau, an evening in memory of Jean-Richard Bloch (the first name is not declined, see § 13, paragraph 3).

3. When declension of Slavic names and surnames, forms of Russian declension are used (in particular, in indirect forms fluent vowels are preserved), for example: Edek, Vladek (Polish names) - Edeka, Vladeka (not “Edka”, “Vladka”); Karel Capek - Karela Capek, (not “Chapka”); Vaclav Havel – Vaclav Havel (not “Gavla”).

4. Russian and foreign surnames ending in a consonant are declined if they refer to men, and not declined if they refer to women. Compare: student Kulik - student Kulik, George Bush - Barbara Bush. Frequent deviations from the rule (indeclinability of Russian male surnames ending in a consonant sound) are observed in cases where the surname is consonant with the name of an animal or inanimate object (Goose, Belt), in order to avoid unusual or curious combinations, for example: “Mr. Goose’s,” "Citizen Belt." Often in such cases, especially in official business speech, keep the last name in initial form(cf.: train with Stanislav Zhuk) or make changes to this type of declension, for example, retain a fluent vowel sound in the forms of oblique cases (cf.: highly appreciate the courage of Konstantin Kobets).

5. Last names are not inclined to -ago, -ako, -yago, -yh, -ikh, -ovo: Shambinago, Plevako, Dubyago, Krasnykh, Dolgikh, Durnovo. Only in common parlance are forms like “Ivan Sedykh’s” found.

6. Foreign surnames ending in a vowel sound (except for unstressed ones) -a, -i, with a preceding consonant) do not decline, for example: the novels of Zola, the poems of Hugo, the operas of Bizet, the music of Puncini, the plays of Shaw, the poems of Salman Rushdie.

Often Slavic (Polish and Czech) surnames are also included under this rule. -ski And -s: opinions of Zbigniew Brzezinski (American social and political figure), Pokorny's dictionary (Czech linguist). It should, however, be borne in mind that the tendency to transfer such surnames in accordance with their sound in the source language (cf. the spelling of the Polish surnames Glinski, Leszczynska - with the letter b before sk) is combined with the tradition of their transmission according to the Russian model in spelling and declension: works by the Polish writer Krasiński, performances by the singer Ewa Bandrowska-Turska, a concert by the pianist Czerna-Stefanska, an article by Octavia Opulska-Danietska, etc. To avoid difficulties in the functioning of such surnames in the Russian language, it is advisable to formalize them according to the model of the declension of Russian male and female surnames into -sky, -tsky, -y, -aya. Polish combinations are inclined in a similar way, for example: Home Army, Home Army, etc.

From surnames to accented ones -A Only the Slavic ones are inclined: From the writer Mayboroda, to the philosopher Skovoroda, the films of Alexander Mitta.

Non-Russian surnames with unstressed names -oh, -i(mainly Slavic and Romanesque) are inclined, for example: the work of Jan Neruda, the poems of Pablo Neruda, the works of honorary academician N.F. Gamaleya, the utopianism of Campanella, the cruelty of Torquemada, the film with the participation of Giulietta Masina; but films starring Henry Fonda and Jane Fonda. Finnish surnames also do not decline to -a: meeting with Kuusela. Foreign surnames do not decline to -ia, for example: sonnets of Heredia, stories of Gulia; on -iya - inclined, for example: the atrocities of Beria.

Fluctuations are observed in the use of Georgian, Japanese and some other surnames; Wed: aria performed by Zurab Sotkilav, songs of Okudzhava, the government of Ardzinba, 100 years since the birth of Saint-Katayama, the politics of General Tanaka, the works of Ryunosuke Akutagawa. IN recent years There has clearly been a tendency towards the decline of such surnames.

7. Ukrainian surnames -ko (-enko) V fiction usually inclined, although different types declensions (as masculine or neuter words), for example: order to the head of Evtukh Makogonenko; poem dedicated to M.V. Rodzianka In modern press, such surnames, as a rule, are not used, for example: the anniversary of Taras Shevchenko, memories of V.G. Korolenko. In some cases, however, their changeability is advisable to add clarity to the text, cf.: letter from V.G. Korolenko A.V. Lunacharsky - letter addressed to V.G. Korolenka. Wed. also from Chekhov: “In the evening Belikov... trudged to Kovalenki.” Surnames are not accentuated: the Franko Theater, Lyashko’s stories.

8. In compound names and surnames of Korean, Vietnamese, Burmese, the last part is declined (if it ends in a consonant), for example: Choi Heng’s speech, Pham Van Dong’s statement, conversation with U Ku Ling.

9. In Russian double surnames, the first part is declined if it is used in itself as a surname, for example: songs by Solovyov-Sedoy, paintings by Sokolov-Skal. If the first part does not form a surname, then it does not decline, for example: research by Grum-Grzhimailo, in the role of Skvoznik-Dmukhanovsky, sculpture by Demut-Malinovsky.

10. Non-Russian surnames referring to two or more persons are in some cases put in the form plural, in others - in the form of a singular:

1) if the surname has two male names, then it is put in the plural form, for example: Heinrich and Thomas Mann, August and Jean Picard, Adolph and Mikhail Gottlieb; also father and son of Oistrakh;
- 2) with two female names, the surname is put in the form singular, for example: Irina and Tamara Press (cf. inflexibility of surnames with a consonant sound related to women);
- 3) if the surname is accompanied by a male and female names, then it retains the singular form, for example: Franklin and Eleanor Roosevelt, Ronald and Nancy Reagan, Ariadne and Peter Tur, Nina and Stanislav Zhuk;
- 4) the surname is also put in the singular if it is accompanied by two common nouns indicating different genders, for example: Mr. and Mrs. Clinton, Lord and Lady Hamilton; however, when combining husband and wife, brother and sister, the surname is more often used in the plural form: husband and wife of Estrema, brother and sister of Niringa;
- 5) when using the word spouse, the surname is given in the singular form, for example: spouse Kent, spouse Major;
- 6) with the word brothers, the surname is also usually put in the singular form, for example: the Grimm brothers, the Spiegel brothers, the Schellenberg brothers, the Pokrass brothers; the same with the word sisters: Koch sisters;
- 7) when using the word family, the surname is usually given in the singular form, for example: Oppenheim family, Hoffmann-Stal family.

11. In combinations of Russian surnames with numerals, the following forms are used: two Petrovs, both Petrovs, two Petrovs, both Petrov brothers, two Petrov friends; two (both) Zhukovskys; two (both) Zhukovskys. This rule also applies to combinations of numerals with foreign-language surnames: both Schlegels, two brothers of Mann.

12. Female middle names Declined according to the type of declension of nouns, and not adjectives, for example: in Anna Ivanovna, to Anna Ivanovna, with Anna Ivanovna.

| § 1.3. Elements of algebra logic

Lessons 8 - 12
§ 1.3. Elements of algebra logic

Key words:

algebra of logic
statement
logical operation
conjunction
disjunction
negation
logical expression
truth table
laws of logic

1.3.1. Statement

Algebra in the broad sense of the word is the science of general operations, similar to addition and multiplication, that can be performed on a variety of mathematical objects. Many mathematical objects (integers and rational numbers, polynomials, vectors, sets) you study in school course algebra, where you get acquainted with such branches of mathematics as the algebra of numbers, the algebra of polynomials, the algebra of sets, etc.

For computer science, a branch of mathematics called logical algebra is important; the objects of the algebra of logic are statements.

An utterance is a sentence in any language whose content can be unambiguously determined to be true or false.

For example, regarding the sentences “The great Russian scientist M.V. Lomonosov was born in 1711” and “Two plus six Is eight” we can definitely say that they are true. The sentence “Sparrows hibernate in winter” is false. Therefore, these sentences are statements.

In Russian, statements are expressed by declarative sentences. But not everything declarative sentence is a statement.

For example, the sentence “This sentence is false” is not a statement, since it cannot be said about it whether it is true or false without obtaining a contradiction. Indeed, if we accept that the sentence is true, then this contradicts what was said. If we accept that the sentence is false, then it follows that it is true.

Regarding the sentence “Computer graphics is the most interesting topic in the course of school computer science”, it is also impossible to say unambiguously whether it is true or false. Think for yourself why.

Incentive and interrogative sentences are not statements.

For example, sentences such as: “Write down homework", "How to get to the library?", "Who came to us?"

Statements can be constructed using signs of various formal languages- mathematics, physics, chemistry, etc.

Examples of statements could be:

“Na is metal” (true statement);
“Newton’s second law is expressed by the formula F=m a” (true statement);
“The perimeter of a rectangle with side lengths a and b is equal to a b” (false statement).

Numerical expressions are not statements, but from two numerical expressions You can make a statement by connecting them with equal or inequality signs. For example:

“3 + 5 = 2 4” (true statement);
“II + VI > VIII” (false statement).

Equalities and inequalities containing variables are also not statements. For example, the sentence "X< 12» становится высказыванием только при замене переменной каким-либо конкретным значением: «5 < 12» - истинное высказывание; «12 < 12» - ложное высказывание.

The justification for the truth or falsity of statements is decided by the sciences to which they belong. The algebra of logic is abstracted from the semantic content of statements. She is only interested in whether a given statement is true or false. In logical algebra, statements are denoted by letters and called logical variables. Moreover, if the statement is true, then the value of the corresponding logical variable is denoted by one (A = 1), and if it is false - by zero (B = 0). 0 and 1 denoting the values of Boolean variables are called Boolean values.

The algebra of logic defines the rules for writing, calculating values, simplifying and transforming statements.

By operating with logical variables, which can only be equal to 0 or 1, the algebra of logic allows you to reduce information processing to operations with binary data. It is the apparatus of logical algebra that forms the basis of computer devices for storing and processing information. You will encounter elements of logical algebra in many other areas of computer science.

1.3.2. Logical operations

Statements can be simple or complex. A statement is called simple if no part of it is itself a statement. Complex (compound) statements are constructed from simple ones using logical operations.

Let's consider the basic logical operations defined on statements. All of them correspond to connectives used in natural language.

Conjunction

Consider two statements: A = “The founder of the algebra of logic is George Boole,” B = “The research of Claude Shannon made it possible to apply the algebra of logic in computer technology.” Obviously, the new statement “The founder of the algebra of logic is George Boole, and the research of Claude Shannon made it possible to apply the algebra of logic in computer technology” is true only if both original statements are true at the same time.

Conjunction is a logical operation that associates each two statements with a new statement, which is true if and only if both original statements are true.

To write a conjunction, the following signs are used: ∧, , И, &. For example: A ∧ B, A B, A AND B, A & B.

The conjunction can be described in the form of a table, which is called a truth table:

The truth table lists all possible values initial statements (columns A and B), and the corresponding binary numbers, as a rule, are arranged in ascending order: 00, 01, 10, 11. The last column records the result of the logical operation for the corresponding operands.

Otherwise, the conjunction is called logical multiplication. Think why.

Disjunction

Consider two statements: A = “The idea of using mathematical symbolism in logic belongs to Gottfried Wilhelm Leibniz,” B = “Leibniz is the founder of binary arithmetic.” Obviously, the new statement “The idea of using mathematical symbolism in logic belongs to Gottfried Wilhelm Leibniz or Leibniz is the founder of binary arithmetic” is false only if both original statements are false at the same time.

Independently determine the truth or falsity of the three statements considered.

Disjunction is a logical operation that associates each two statements with a new statement, which is false if and only if both original statements are false.

To write a disjunction, the following signs are used: ∨, |, OR, +. For example: A∨B, A|B, A OR B, A+B.

The disjunction is defined by the following truth table:

Otherwise, disjunction is called logical addition. Think why.

Inversion

Inversion is a logical operation that associates each statement with a new statement, the meaning of which is opposite to the original one.

To write inversion, the following signs are used: NOT, ¬, ‾. For example: NOT A, ¬A, .

The inversion is determined by the following truth table:

Inversion is otherwise called logical negation.

The negation of the statement “I have a computer at home” will be the statement “It is not true that I have a computer at home” or, which is the same in Russian, “I do not have a computer at home.” Denial of the statement “I don’t know” Chinese” will be the statement “It is not true that I do not know Chinese” or, which is the same thing in Russian, “I know Chinese.” The negation of the statement “All 9th grade boys are excellent students” is the statement “It is not true that all 9th grade boys are excellent students,” in other words, “Not all 9th grade boys are excellent students.”

Thus, when constructing a negation to a simple statement, either the phrase “it is not true that...” is used, or the negation is constructed to a predicate, then the particle “not” is added to the corresponding verb.

Any complex statement can be written as a logical expression - an expression containing logical variables, logical operator signs and parentheses. Logical operations in a logical expression are performed in the following order: inversion, conjunction, disjunction. You can change the order of operations using parentheses.

Logical operations have the following priority: inversion, conjunction, disjunction.

Example 1 . Let A = “The word “cruiser” appears on the Web page,” B = “The word “battleship” appears on the Web page.” We are considering a certain segment of the Internet containing 5,000,000 Web pages. In it, statement A is true for 4800 pages, statement B is true for 4500 pages, and statement A v B is true for 7000 pages. For how many Web pages will the following expressions and statements be true in this case?

a) NOT (A OR B);

c) The word “cruiser” appears on the Web page, but the word “battleship” does not appear.

Solution . Let us depict the set of all Web pages of the Internet sector under consideration as a circle, inside which we will place two circles: one of them corresponds to the set of Web pages where statement A is true, the second - where statement B is true (Fig. 1.3).

Rice. 1.3.
Graphic representation of multiple Web pages

Let's graphically depict the sets of Web pages for which expressions and statements a) - c) are true (Fig. 1.4)

Rice. 1.4.
Graphic representation of sets of Web pages for which expressions and statements a) - c) are true

The constructed diagrams will help us answer the questions contained in the task.

The expression A OR B is true for 7,000 Web pages, and there are 5,000,000 pages in total. Therefore, the expression A OR B is false for 4,993,000 Web pages. In other words, for 4,993,000 Web pages, the expression NOT (A OR B) is true.

The expression A ∨ B is true for those Web pages where A is true (4800), as well as for those Web pages where B is true (4500). If all Web pages were different, then the expression A v B would be true for 9300 (4800 + 4500) Web pages. But, according to the condition, there are only 7000 such Web pages. This means that on 2300 (9300 - 7000) Web pages both words appear simultaneously. Therefore, expression A & B is true for 2300 Web pages.

To find out for how many Web pages statement A is true and at the same time statement B is false, subtract 2300 from 4800. Thus, the statement “The word “cruiser” appears on the Web page and the word “battleship” does not appear” is true on 2500 Web pages. pages.

Write down the logical expression corresponding to the statement considered.

On the website Federal Center information and educational resources (http://fcoir.edu.ru/) contains the information module “Statement. Simple and complex statements. Basic logical operations". Getting to know this resource will allow you to expand your understanding of the topic you are studying.

1.3.3. Construction of truth tables for logical expressions

For a logical expression, you can build a truth table showing what values the expression takes for all sets of values of the variables included in it. To construct a truth table you should:

count n - the number of variables in the expression;
count total number logical operations in expression;
establish the sequence of logical operations, taking into account parentheses and priorities;
determine the number of columns in the table: number of variables + number of operations;
fill out the header of the table, including variables and operations in accordance with the sequence established in paragraph 3;
determine the number of rows in the table (not counting the table header) m = 2n;
write down sets of input variables, taking into account the fact that they represent a whole series of n-bit binary numbers from 0 to 2 n - 1;
fill the table column by column, performing logical operations in accordance with the established sequence.

Let's build a truth table for the logical expression A ∨ A & B. It contains two variables, two operations, and first the conjunction is performed, and then the disjunction. The table will have four columns in total:

Sets of input variables are integers from O to 3, presented in two-digit binary code: 00, 01, 10, 11. The completed truth table looks like:

Note that the last column (result) is the same as column A. In this case, the logical expression A ∨ A & B is said to be equivalent to the logical expression A.

1.3.4. Properties of logical operations

Let's consider the basic properties (laws) of the algebra of logic.

Commutative (commutative) law

for logical multiplication:

A & B = B & A;

for logical addition:

A ∨ B = B ∨ A.

Combinative (associative) law

for logical multiplication:

(A & B) & C = A & (B & C);

for logical addition:

(A ∨ B) ∨ C = A ∨(B ∨ C).

If the signs of the operations are the same, the parentheses can be placed arbitrarily or omitted altogether.

Distributive (distributive) law

for logical multiplication:

A & (B ∨ C) = (A & B) ∨ (A & C);

for logical addition:

A ∨ (B & C) = (A ∨ B) & (A ∨ C).

Law of double negation

Law of exclusion of middle

Of two contradictory statements about the same subject, one is always true, the second is false, and there is no third.

Law of Repetition

for logical multiplication:
for logical addition:

Laws of operations with 0 and 1

for logical multiplication:

A & 0 = 0; A & 1 = A;

for logical addition:

A ∨ O = A; A ∨ l = l.

Laws of general inversion

The laws of logical algebra can be proven using truth tables.

Let us prove the distribution law for logical addition:

A ∨ (B & C) = (A ∨ B) & (A ∨ C).

The coincidence of the columns corresponding to the logical expressions on the left and right sides of the equality proves the validity of the distribution law for logical addition.

Example 2 . Let's find the value of a logical expression for the number X = 0.

Solution . When X = 0 we obtain the following logical expression: . Since logical expressions are 0< 3, 0 < 2 истинны, то, подставив их значения в логическое выражение, получаем: 1&Т = 1&0 = 0.

1.3.5. Solving logical problems

Let's consider several solutions logical problems.

Problem 1 . Kolya, Vasya and Seryozha were visiting their grandmother in the summer. One day one of the boys accidentally broke his grandmother's favorite vase. When asked who broke the vase, they gave the following answers:

Seryozha: 1) I didn’t break it. 2) Vasya didn’t break it.

Vasya: 3) Seryozha didn’t break it. 4) Kolya broke the vase.

Kolya: 5) I didn’t break it. 6) Seryozha broke the vase.

The grandmother knew that one of her grandchildren, let's call him truthful, told the truth both times; the second, let's call him a joker, told a lie both times; the third, let's call him a cunning one, told the truth once, and another time - a lie. Name the truthful, the joker and the cunning. Which grandson broke the vase?

Solution. Let K = “Kolya broke a vase”, B = “Vasya broke a vase”, C = “Seryozha broke a vase”. Let's make a truth table with which we present the statements of each boy 1 .

1 Taking into account the fact that the vase was broken by one grandson, it was possible to create not the entire table, but only its fragment containing the following sets of input variables: 001, 010, 100.

Based on what the grandmother knows about her grandchildren, you should look for rows in the table that contain, in any order, three combinations of values: 00, 11, 01 (or 10). There were two such rows in the table (they are marked with check marks). According to the second of them, the vase was broken by Kolya and Vasya, which contradicts the condition. According to the first of the lines found, Seryozha broke the vase, and he turned out to be a cunning one. Vasya turned out to be the joker. The name of the truthful grandson is Kolya.

Problem 2 . Alla, Valya, Sima and Dasha are participating in gymnastics competitions. Fans made suggestions about possible winners:

Sima will be first, Valya will be second;
Sima will be second, Dasha will be third;
Alla will be second, Dasha will be fourth.

At the end of the competition, it turned out that in each of the assumptions only one of the statements is true, the other is false. What place did each of the girls take in the competition if they all ended up in different places?

Solution . Let's look at some simple statements:

C 1 = “Sima took first place”;

B 2 = “Valya took second place”;

C 2 = “Sima took second place”;

D 3 = “Dasha took third place”;

A 2 = “Alla took second place”;

D 4 = “Dasha took fourth place.”

Since in each of the three assumptions one of the statements is true and the other is false, we can conclude the following:

C 1 + B 2 = 1, C 1 B 2 = 0;
C 2 + D 3 = 1, C 2 D 3 = 0;
A 2 + D 4 = 1, A 2 D 4 = 0.

The logical product of true statements will be true:

(C 1 + B 2) (C 2 + D 3) (A 2 + D 4) = 1.

Based on the distribution law, we transform the left side of this expression:

(C 1 C 2 + C 1 D 3 + B 2 C 2 + B 2 D 3) (A 2 + D 4) = 1.

The statement C 1 C 2 means that Sima took both first and second places. According to the conditions of the problem, this statement is false. The statement B 2 C 2 is also false. Taking into account the law of operations with the constant 0, we write:

(C 1 D 3 + B 2 D 3) (A 2 + D 4) = 1.

Further transformation of the left side of this equality and exclusion of obviously false statements gives:

C 1 D 3 A 2 + C 1 D 3 D 4 + B 2 D 3 A 2 + B 2 D 3 D 4 = 1.

C 1 D 3 A 2 = 1.

From the last equality it follows that C 1 = 1, D 3 = 1, A 2 = 1. This means that Sima took first place, Alla took second, Dasha took third. Consequently, Valya took fourth place.

You can get acquainted with other ways to solve logical problems, as well as take part in Internet Olympiads and competitions for solving them on the website “Mathematics for Schoolchildren” (http://www.kenqyry.com/).

On the website http://www.kaser.com/ you can download a demo version of a very useful one that develops logic and reasoning skills logic puzzle Sherlock.

1.3.6. Logic elements

Algebra of logic is a branch of mathematics that plays an important role in the design of automatic devices and the development of hardware and software for information and communication technologies.

You already know that any information can be presented in discrete form- in the form of a fixed set of individual values. Devices that process such values (signals) are called discrete. A discrete converter that, after processing binary signals, produces the value of one of the logical operations is called a logical element.

In Fig. 1.5 are given symbols(circuits) of logical elements that implement logical multiplication, logical addition and inversion.

Figure 1.5.
Logic elements

The AND logical element (conjunctor) implements the logical multiplication operation (Fig. 1.5, a). A unit at the output of this element will appear only when there are units at all inputs.

The OR logical element (disjunctor) implements the logical addition operation (Fig. 1.5, b). If at least one input is one, then the output of the element will also be one.

The NOT logical element (inverter) implements the negation operation (Fig. 1.5, c). If the input of the element is O, then the output is 1 and vice versa.

Computer devices that perform operations on binary numbers, and the cells that store data are electronic circuits consisting of individual logic elements. These issues will be covered in more detail in the computer science course for grades 10-11.

Example 3. Let's analyze the electronic circuit, that is, find out what signal should be at the output for each possible set of signals at the inputs.

Solution. We will enter all possible combinations of signals at inputs A to B into the truth table. Let's trace the transformation of each pair of signals as they pass through logical elements and write the result in a table. The completed truth table completely describes the electronic circuit under consideration.

A truth table can also be constructed using a logical expression corresponding to an electronic circuit. The last logical element in the circuit under consideration is the conjunctor. It receives signals from input L and from the inverter. In turn, the inverter receives a signal from input B. Thus,

Gain a more complete understanding of logical elements and electronic circuits Working with the Logic simulator (http://kpolyakov.narod.ru/prog/logic.htm) will help you.

The most important

Statement is a sentence in any language, the content of which can be unambiguously determined as true or false.

Basic logical operations defined on statements: inversion, conjunction, disjunction.

Truth tables for basic logical operations:

When evaluating Boolean expressions, the steps in parentheses are performed first. Priority of execution of logical operations:

Questions and tasks

Explain why the following sentences are not statements.

What color is this house?
The number X does not exceed one.
4X + 3.
Look out the window.
Drink tomato juice!
This topic is boring.
Ricky Martin is the most popular singer.
Have you been to the theater?

Give one example of true and false statements from biology, geography, computer science, history, mathematics, literature.
In the following statements, highlight the simple statements, indicating each of them with a letter; write down each compound statement using letters and logical operations signs.

The number 376 is even and has three digits.
In winter, children go ice skating or skiing.
We will celebrate the New Year at the dacha or on Red Square.
It is not true that the Sun moves around the Earth.
The Earth is shaped like a ball, which appears blue from space.
During the mathematics lesson, high school students answered the teacher’s questions and also wrote independent work.

Construct the negation of the following statements.

Today the opera “Eugene Onegin” is being performed at the theater.
Every hunter wants to know where the pheasant is sitting.
The number 1 is a prime number.
Natural numbers ending in 0 are not prime numbers.
It is not true that the number 3 is not a divisor of the number 198.
Kolya solved all the tasks of the test.
At every school, some students are interested in sports.
Some mammals do not live on land.

Let A = “Anya likes math lessons,” and B = “Anya likes chemistry lessons.” Express the following formulas in ordinary language:

Consider the electrical circuits shown in the figure:

They depict the parallel and serial connections switches. In the first case, both switches must be turned on for the light to light up. In the second case, it is enough that one of the switches is turned on. Try to draw an analogy between the elements yourself electrical diagrams and objects and operations of logical algebra:

Some segment of the Internet consists of 1000 sites. Search server in automatic mode compiled a table of keywords for sites in this segment. Here is its fragment:

The query catfish & guppies found 0 sites, the query catfish & swordtails found 20 sites, and the query swordtails & guppies found 10 sites.

How many sites will be found for the query catfish | swordtails | guppy?

For how many sites in the segment under consideration is the statement “Catfish - keyword site OR swordtails - the keyword of the site OR guppies - the keyword of the site?

Construct truth tables for the following logical expressions:

Conduct a proof of the logical laws discussed in the paragraph using truth tables.
Given three numbers in the decimal number system: A = 23, B = 19, C = 26. Convert A, B and C to the binary number system and perform bitwise logical operations (A ∨ B) & C. Give the answer in the decimal number system.
Find the meanings of the expressions:

Find the value of a Boolean expression for the specified values of the number X:

1) 1;
2) 2;
3) 3;
4) 4

Let A = “The first letter of the name is a vowel”, B = “The fourth letter of the name is a consonant”. Find the value of the Boolean expression for the following names:

4) FEDOR

The case of John, Brown and Smith is being examined. It is known that one of them found and hid the treasure. During the investigation, each of the suspects made two statements:

Smith: “I didn’t do it. Brown did it."

John: Brown is not guilty. Smith did it."

Brown: “I didn’t do it. John didn't do it."

The court found that one of them lied twice, the other told the truth twice, the third lied once and told the truth once. Which suspect should be acquitted?

Alyosha, Borya and Grisha found an ancient vessel in the ground. Examining the amazing find, each made two assumptions:
1. Alyosha: “This is a Greek vessel and was made in the 5th century.”
2. Borya: “This is a Phoenician vessel and was made in the 3rd century.”
3. Grisha: “This vessel is not Greek and was made in the 4th century.”
  The history teacher told the children that each of them was right in only one of two assumptions. Where and in what century was the vessel made?
Find out what signal should be at the output of the electronic circuit for each possible set of signals at the inputs. Make a table of how the circuit works. What logical expression describes the circuit?

Key words:

algebra of logic
statement
logical operation
conjunction
disjunction
negation
logical expression
truth table
laws of logic

1.3.1. Statement

Algebra in the broad sense of the word is the science of general operations, similar to addition and multiplication, that can be performed on a variety of mathematical objects. You study many mathematical objects (integer and rational numbers, polynomials, vectors, sets) in a school algebra course, where you become familiar with such branches of mathematics as the algebra of numbers, the algebra of polynomials, the algebra of sets, etc.

For computer science, a branch of mathematics called logical algebra is important; The objects of the algebra of logic are statements.

For example, regarding the sentences “The great Russian scientist M.V. Lomonosov was born in 1711” and “Two plus six Is eight” we can definitely say that they are true. The sentence “Sparrows hibernate in winter” is false. Therefore, these sentences are statements.

For example, the sentence “This sentence is false” is not a statement because it cannot be said whether it is true or false without obtaining a contradiction. Indeed, if we accept that the sentence is true, then this contradicts what was said. If we accept that the sentence is false, then it follows that it is true.

Regarding the sentence “Computer graphics is the most interesting topic in the school computer science course,” it is also impossible to say unambiguously whether it is true or false. Think for yourself why.

For example, sentences such as: “Write down your homework”, “How to get to the library?”, “Who came to us?” are not statements. "

Examples of statements could be:

“Na is metal” (true statement);
“Newton’s second law is expressed by the formula F=m a” (true statement);
“The perimeter of a rectangle with side lengths a u b is equal to a b” (false statement).

Numerical expressions are not statements, but from two numerical expressions you can make a statement by connecting them with equal or inequality signs. For example:

“34-5 = 2 4” (true statement);
“II4-VI > VIII” (false statement).

1.3.2. Logical operations

Statements can be simple or complex. A statement is called simple if no part of it is itself a statement. Complex (compound) statements are constructed from simple ones using logical operations.

Let's consider the basic logical operations defined on statements. All of them correspond to connectives used in natural language.

Conjunction

To write a conjunction, the following signs are used: , , И, &. For example: A B, A B, A AND B, A&B.

The conjunction can be described in the form of a table, which is called a truth table:

The truth table lists all possible values of the original statements (columns A and B), and the corresponding binary numbers are usually arranged in ascending order: 00, 01, 10, 11. The last column records the result of the logical operation for the corresponding operands.

Otherwise, the conjunction is called logical multiplication. Think why.

Disjunction

Independently determine the truth or falsity of the three statements considered.

To write a disjunction, the following signs are used: v, |, OR, +. For example: AvB, A|B, A OR B, A+B.

The disjunction is defined by the following truth table:

Otherwise, disjunction is called logical addition. Think why.

Inversion

To write inversion, the following signs are used: NOT, ¬, ‾. For example: NOT, ¬, ‾.

The inversion is determined by the following truth table:

Inversion is otherwise called logical negation.

The negation of the statement “I have a computer at home” will be the statement “It is not true that I have a computer at home” or, which is the same in Russian, “I do not have a computer at home.” The negation of the statement “I don’t know Chinese” will be the statement “It is not true that I don’t know Chinese” or, which is the same thing in Russian, “I know Chinese.” The negation of the statement “All 9th grade boys are excellent students” is the statement “It is not true that all 9th grade boys are excellent students,” in other words, “Not all 9th grade boys are excellent students.”

Example 1. Let A = “The word “cruiser” appears on the Web page,” B = “The word “battleship” appears on the Web page.” We are considering a certain segment of the Internet containing 5,000,000 Web pages. In it, statement A is true for 4800 pages, statement B is true for 4500 pages, and statement A v B is true for 7000 pages. For how many Web pages will the following expressions and statements be true in this case?

a) NOT (A OR B);

c) The word “cruiser” appears on the Web page, but the word “battleship” does not appear.

Solution. Let us depict the set of all Web pages of the Internet sector under consideration as a circle, inside which we will place two circles: one of them corresponds to the set of Web pages where statement A is true, the second - where statement B is true (Fig. 1.3).

Rice. 1.3.
Graphic representation of multiple Web pages

Let's graphically depict the sets of Web pages for which expressions and statements a) - c) are true (Fig. 1.4)

Rice. 1.4.
Graphic representation of sets of Web pages for which expressions and statements a) - c) are true

The constructed diagrams will help us answer the questions contained in the task.

The expression A v B is true for those Web pages where A (4800) is true, as well as those Web pages where B (4500) is true. If all Web pages were different, then the expression A v B would be true for 9300 (4800 + 4500) Web pages. But, according to the condition, there are only 7000 such Web pages. This means that on 2300 (9300 - 7000) Web pages both words appear simultaneously. Therefore, expression A & B is true for 2300 Web pages.

Write down the logical expression corresponding to the statement considered.

The website of the Federal Center for Information and Educational Resources (http://fcoir.edu.ru/) contains the information module “Statement. Simple and complex statements. Basic logical operations". Getting to know this resource will allow you to expand your understanding of the topic you are studying.

1.3.3. Construction of truth tables for logical expressions

For a logical expression, you can build a truth table showing what values the expression takes for all sets of values of the variables included in it. To construct a truth table you should:

count n - the number of variables in the expression;
count the total number of logical operations in an expression;
establish the sequence of logical operations, taking into account parentheses and priorities;
determine the number of columns in the table: number of variables + number of operations;
fill out the header of the table, including variables and operations in accordance with the sequence established in paragraph 3;
determine the number of rows in the table (not counting the table header) m = 2n;
write down sets of input variables, taking into account the fact that they represent a whole series of n-bit binary numbers from 0 to 2 n - 1;
fill the table column by column, performing logical operations in accordance with the established sequence.

Let's build a truth table for the logical expression A v A & B. It contains two variables, two operations, and first the conjunction is performed, and then the disjunction. The table will have four columns in total:

Sets of input variables are integers from O to 3, presented in two-digit binary code: 00, 01, 10, 11. The completed truth table looks like:

Note that the last column (result) is the same as column A. In this case, the logical expression A v A & B is said to be equivalent to the logical expression A.

1.3.4. Properties of logical operations

Let's consider the basic properties (laws) of the algebra of logic.

The laws of logical algebra can be proven using truth tables.

Let us prove the distribution law for logical addition:

A v (B & C) = (A V B) & (A v C).

The coincidence of the columns corresponding to the logical expressions on the left and right sides of the equality proves the validity of the distribution law for logical addition.

Example 2. Let's find the value of a logical expression for the number X = 0.

Solution. When X = 0 we obtain the following logical expression: . Since logical expressions are 0< 3, 0 < 2 истинны, то, подставив их значения в логическое выражение, получаем: 1&Т = 1&0 = 0.

1.3.5. Solving logical problems

Let's look at several ways to solve logical problems.

Problem 1. Kolya, Vasya and Seryozha were visiting their grandmother in the summer. One day one of the boys accidentally broke his grandmother's favorite vase. When asked who broke the vase, they gave the following answers:

Seryozha: 1) I didn’t break it. 2) Vasya didn’t break it.

Vasya: 3) Seryozha didn’t break it. 4) Kolya broke the vase.

Kolya: 5) I didn’t break it. 6) Seryozha broke the vase.

Solution. Let K = “Kolya broke a vase”, B = “Vasya broke a vase”, C = “Seryozha broke a vase”. Let's create a truth table with which we present the statements of each boy 1.

Problem 2. Alla, Valya, Sima and Dasha are participating in gymnastics competitions. Fans made suggestions about possible winners:

Sima will be first, Valya will be second;
Sima will be second, Dasha will be third;
Alla will be second, Dasha will be fourth.

Solution. Let's look at some simple statements:

C 1 = “Sima took first place”;

B 2 = “Valya took second place”;

C 2 = “Sima took second place”;

D 3 = “Dasha took third place”;

A 2 = “Alla took second place”;

D 4 = “Dasha took fourth place.”

Since in each of the three assumptions one of the statements is true and the other is false, we can conclude the following:

C 1 + B 2 = 1, C 1 B 2 = 0;
C 2 + D 3 = 1, C 2 D 3 = 0;
A 2 + D 4 = 1, A 2 D 4 = 0.

The logical product of true statements will be true:

(C 1 + B 2) (C 2 + D 3) (A 2 + D 4) = 1.

Based on the distribution law, we transform the left side of this expression:

(C 1 C 2 + C 1 D 3 + B 2 C 2 + B 2 D 3) (A 2 + D 4) = 1.

(C 1 D 3 + B 2 D 3) (A 2 + D 4) = 1.

Further transformation of the left side of this equality and exclusion of obviously false statements gives:

C 1 D 3 A 2 + C 1 D 3 D 4 + B 2 D 3 A 2 + B 2 D 3 D 4 = 1.

C 1 D 3 A 2 = 1.

From the last equality it follows that C 1 = 1, D 3 = 1, A 2 = 1. This means that Sima took first place, Alla took second, Dasha took third. Consequently, Valya took fourth place.

On the website http://www.kaser.com/ you can download a demo version of a very useful Sherlock logic puzzle that develops logic and reasoning skills.

1.3.6. Logic elements

You already know that any information can be represented in discrete form - as a fixed set of individual values. Devices that process such values (signals) are called discrete. A discrete converter that, after processing binary signals, produces the value of one of the logical operations is called a logical element.

In Fig. 1.5 shows the symbols (diagrams) of logical elements that implement logical multiplication, logical addition and inversion.

Figure 1.5.
Logic elements

The AND logical element (conjunctor) implements the logical multiplication operation (Fig. 1.5, a). A unit at the output of this element will appear only when there are units at all inputs.

The OR logical element (disjunctor) implements the logical addition operation (Fig. 1.5, b). If at least one input is one, then the output of the element will also be one.

The NOT logical element (inverter) implements the negation operation (Fig. 1.5, c). If the input of the element is O, then the output is 1 and vice versa.

Computer devices that perform operations on binary numbers and cells that store data are electronic circuits consisting of individual logical elements. These issues will be covered in more detail in the computer science course for grades 10-11.

Example 3. Let's analyze the electronic circuit, that is, find out what signal should be at the output for each possible set of signals at the inputs.

Working with the Logic simulator (http://kpolyakov. narod. ru/prog/logic. htm) will help you get a more complete understanding of logical elements and electronic circuits.