The concept of “utterance” is primary. In logic, a statement is a declarative sentence that can be said to be true or false. Every statement is either true or false, and no statement is both true and false.
Examples of statements: there is an even number”, “1 is a prime number”. The truth value of the first two statements is “truth”, the truth value of the last two
Interrogative and exclamatory sentences are not statements. Definitions are not statements. For example, the definition “an integer is said to be even if it is divisible by 2” is not a statement. However, the declarative sentence “if an integer is divisible by 2, then it is even” is a statement, and a true one at that. In propositional logic, one abstracts from the semantic content of a statement, limiting itself to considering it from the position that it is either true or false.
In what follows, we will understand the meaning of a statement as its truth value (“true” or “false”). We will denote statements in capital Latin letters, and their meanings, i.e. “true” or “false,” by the letters I and L, respectively.
Propositional logic studies connections that are completely determined by the way in which some statements are built from others, called elementary ones. In this case, elementary statements are considered as wholes, not decomposable into parts, the internal structure of which will not interest us.
Logical operations on statements.
From elementary statements, using logical operations, you can obtain new, more complex statements. The truth value of a complex statement depends on the truth values of the statements that make up the complex statement. This dependence is established in the definitions below and is reflected in the truth tables. The left columns of these tables contain all possible distributions of truth values for statements that directly constitute the complex statement under consideration. In the right column, write the truth values of the complex statement according to the distributions in each row.
Let A and B be arbitrary statements about which we do not assume that their truth values are known. The negation of a statement A is a new statement that is true if and only if A is false. The negation of A is indicated by and reads “not A” or “it is not true that A.” The negation operation is completely determined by the truth table
Example. The statement “it is not true that 5 is an even number,” which has the value I, is the negation of the false statement “5 is an even number.”
Using the operation of conjunction, two statements are formed into one complex statement, denoted A D B. By definition, the statement A D B is true if and only if both statements are true. Statements A and B are called, respectively, the first and second members of the conjunction A D B. The entry “A D B” is read as “L and B”. The truth table for the conjunction has the form
Example. The statement “7 is a prime number and 6 is an odd number” is false as a conjunction of two statements, one of which is false.
The disjunction of two statements A and B is a statement, denoted by , that is true if and only if at least one of the statements A and B is true.
Accordingly, the statement A V B is false if and only if both A and B are false. Statements A and B are called, respectively, the first and second terms of the disjunction A V B. The entry A V B is read as “A or B.” The conjunction "or" in in this case has an inseparable meaning, since the statement A V B is true even if both terms are true. The disjunction has the following truth table:
Example. Statement “3 A statement, denoted by , is false if and only if A is true and B is false, is called an implication with premise A and conclusion B. The statement A-+ B is read as “if A, then 5,” or “ A implies B,” or “from A follows B.” The truth table for the implication is:
Note that there may be no cause-and-effect relationship between the premise and the conclusion, but this cannot affect the truth or falsity of the implication. For example, the statement “if 5 is a prime number, then the bisector of an equilateral triangle is the median” will be true, although in the usual sense the second does not follow from the first. The statement “if 2 + 2 = 5, then 6 + 3 = 9” will also be true, since its conclusion is true. At this definition, if the conclusion is true, the implication will be true regardless of the truth value of the premise. When the premise is false, the implication will be true regardless of the truth value of the conclusion. These circumstances are briefly formulated as follows: “truth follows from anything,” “anything follows from false.”
Various judgments can be made regarding concepts and relationships between them. The linguistic form of judgments is narrative sentences. Sentences used in mathematics can be written either verbally or symbolically. Sentences may contain true or false information.
By saying is any declarative sentence that can be either true or false.
Example. The following sentences are propositions:
1) All MSPU students are excellent students (false statement),
2) There are crocodiles on the Kola Peninsula (false statement),
3) The diagonals of the rectangle are equal (true statement),
4) The equation has no real roots (true statement),
5) The number 21 is even (false statement).
The following sentences are not statements:
What the weather will be tomorrow?
X- natural number,
745 + 231 – 64.
Statements are usually denoted in capital letters of the Latin alphabet: A, B, C,…,Z.
"True" and "falsehood" are called truth values of a statement . Every statement is either true or false; it cannot be both at the same time.
Record [ A ] = 1 means that the statement A – true .
And the recording [ A ] = 0 means that the statement A – false .
Offer 
is not a statement, since it is impossible to say about it whether it is true or false. When substituting specific values for a variable X it turns into a statement: true or false.
Example. If 
, That 
- a false statement, and if 
, That 
- true statement.
Offer 
called predicate or expressive form. It generates many statements of the same form.
Predicate is a sentence with one or more variables that turns into a statement whenever their values are substituted for the variables.
Depending on the number of variables included in the offer, there are single, double, triple, etc. predicates that are denoted by: etc.
Example.
1)
– one-place predicate,
2) "Direct" X perpendicular to a straight line at" is a two-place predicate.
Predicates can also contain variables implicitly. In the sentences: “The number is even”, “two lines intersect” there are no variables, but they are implied: “The number X– even”, “two straight X And at intersect."
When specifying a predicate, indicate it domain – a set from which the values of the variables included in the predicate are selected.
Example. Inequality 
can be considered on a set natural numbers, but we can assume that the value of the variable is selected from the set of real numbers. In the first case, the domain of definition of the inequality 
there will be a set of natural numbers, and in the second - a set of real numbers.
One-place predicate , defined on the set X, is a sentence with a variable that turns into a statement when a variable from the set is substituted into it X.
The set of truth A one-place predicate is the set of those values of a variable from the domain of its definition, upon substitution of which the predicate turns into a true statement.
Example. The truth set of a predicate 
, given on the set of real numbers, there will be an interval 
. Truth set of a predicate 
, defined on the set of non-negative integers, consists of one number 2.
Truth set
two-place predicate 
consists of all such pairs 
when substituted into this predicate, a true statement is obtained.
Example. Pair 
belongs to the truth set of the predicate 
, because 
is a true statement, and the pair 
does not belong, because 
- a false statement.
Statements and predicates can be either simple or complex (composite). Complex sentences are formed from simple ones using logical connectives – words “ And », « or », « if... then … », « then and only then when... » . Using a particle « Not » or phrases " it is not true that » possible from this proposal get new. Sentences that are not compound are called elementary .
Examples. Compound sentences:
The number 42 is even and is divisible by 7. It is formed from two elementary sentences: The number 42 is even, the number 42 is divisible by 7 and is composed using the logical connective “ And ».
Number X greater than or equal to 5. Formed from two elementary sentences: Number X more than 5 and number X equals 5 and is composed using the logical connective " or ».
The number 42 is not divisible by 5. Formed from the sentence: The number 42 is divisible by 5 using the particle “ Not ».
The truth value of an elementary statement is determined based on its content based on known knowledge. To determine the truth value of a compound statement, you need to know the meaning of the logical connectives with the help of which it is formed from elementary ones, and be able to identify the logical structure of the statement.
Example. Let us identify the logical structure of the sentence: “If the angles are vertical, then they are equal.” It consists of two elementary sentences: A– vertical angles, IN- the angles are equal. They are connected into one compound sentence using a logical connective “ if... then..." This compound sentence has the logical structure (form): “ if A, then IN».
The expression "for anyone" X" or "for everyone X" or "for everyone X"is called general quantifier
and is designated 
.
using a general quantifier, it is denoted: 
and reads: “For any value X from many X occurs 
».
The expression "there is X" or "for some X"or "there will be such X"is called existence quantifier
and is designated 
.
Statement derived from a proposition or predicate 
using the existence quantifier, it is denoted: 
and reads: “For some X from many X occurs 
" or "There is (there is) such a meaning X from X what is happening 
».
Quantifiers of generality and existence are used not only in mathematical expressions, but also in everyday speech.
Example. The following statements contain a general quantifier:
a) All sides of the square are equal; b) Every integer is real; c) In any triangle, the medians intersect at one point; d) All students have a grade book.
The following statements contain an existence quantifier:
a) There are numbers that are multiples of 5; b) There is such a natural number 
, What 
; c) Some student groups include candidates for master of sports; d) At least one angle in the triangle is acute.
Statement 
is true
–
identity, i.e. takes true values when any variable values are substituted into it.
Example. Statement 
true.
Statement 
false
, if for some value of the variable X predicate 
Example. Statement 
false, because at 
predicate 
turns into a false statement.
Statement 
is true
if and only if the predicate 
is not identically false, i.e. at some value of the variable X predicate 
Example. Statement 
true, because at 
predicate 
turns into a true statement.
Statement 
false
, if the predicate 
is a contradiction, i.e. identically a false statement.
Example. Statement 
false, because predicate 
is identically false.
Let the offer A - statement. If you put the particle “ Not
"or before the entire sentence put the words " it is not true that
", then we get a new sentence called denial given and is denoted:
A or 
(read: " Not
A" or " it is not true that
A
»).
|
Negation of statement A
called a statement Negation truth table: |
|
or
A, which is false when the statement A true, and true when the statement A– false.
Example. If the statement A: “Vertical angles are equal,” then the negation of this statement A: "The vertical angles are not equal." The first of these statements is true, and the second is false.
To construct the negation of statements with quantifiers you need:
replace the quantifier of generality with the quantifier of existence or vice versa;
replace the statement with its negation (put the particle “ Not»).
On the tongue mathematical symbols it will be written like this.
Plan
Statements with external negation.
Conjunctive statements.
Disjunctive statements.
Strictly disjunctive statements.
Equivalence statements.
Implicative statements.
Statements with external negation.
A statement with external negation is a statement (judgment) that asserts the absence of a certain situation. It is most often expressed by a sentence beginning with the phrase “it is not true that...” or “it is not correct that...”. External negation is indicated by the symbol “ù”, called the negation sign. This sign is determined by the following truth table:
In statements with external negation, the situation in A is denied. For example, if A: “The Volga flows into the Black Sea,” then ùA: “It is not true that the Volga flows into the Black Sea.”
Conjunctive statements.
Conjunctive statements are those that state the simultaneous presence of two situations. Conjunctive statements are formed from two statements using the conjunctions “and”, “a”, “but”. Form of conjunctive statement: (A&B). Each of the statements A and B can take on either the value “true” or the value “false”. For brevity, these values are denoted by the letters i, l. The truth table for conjunctive statements is as follows:
Conjunctive statements state that the situation described in A and in B occur simultaneously. Examples of conjunctive statements: “The Earth is a planet, and the Moon is a satellite”; “Petrov mastered logic well, but Sidorov mastered logic poorly”; “It’s dark outside, and the lights are on in the classroom”; “Petrov bribed the official with money, and Sidorov with a bottle.”
Disjunctive statements.
Disjunctive statements are statements that assert the existence of at least one of the two situations described in A and B. A disjunction is denoted by the symbol V and is expressed in natural language by the conjunction “or.”
The tabular definition of the disjunction sign has the following form:
An example of a disjunctive statement: “Roman Sergeevich Ivanov is a teacher, or Roman Sergeevich Ivanov is a graduate student.”
Strictly disjunctive statements.
Strictly disjunctive statements are statements that assert the presence of exactly one of the two situations described in A and B. Such statements are most often made through sentences with the conjunction “or..., or...” (“either... or ..."). A strict disjunction is denoted by the symbol V* (read “either... or...").
The tabular definition of the strict disjunction sign has the following form:
An example of a strictly disjunctive statement: “It’s either sunny outside or it’s raining.”
Properties
Let's consider several properties of the Cartesian product:
1. If A,B are finite sets, then A× B- final. And vice versa, if one of the factor sets is infinite, then the result of their product is an infinite set.
2. The number of elements in a Cartesian product is equal to the product of the numbers of elements of the factor sets (if they are finite, of course): | A× B|=|A|⋅|B| .
3. A np ≠(A n) p- in the first case, it is advisable to consider the result of the Cartesian product as a matrix of dimensions 1× n.p., in the second - as a matrix of sizes n× p .
4. The commutative law is not satisfied, because pairs of elements of the result of a Cartesian product are ordered: A× B≠B× A .
5. The associative law is not fulfilled: ( A× B)× C≠A×( B× C) .
6. There is distributivity with respect to basic operations on sets: ( A∗B)× C=(A× C)∗(B× C),∗∈{∩,∪,∖}
11. The concept of a statement. Elementary and compound statements.
Statement is a statement or declarative sentence that can be said to be true (I-1) or false (F-0), but not both.
For example, "Today it's raining", "Ivanov completed laboratory work No. 2 in physics."
If we have several initial statements, then from them, using logical unions or particles we can form new statements, the truth value of which depends only on the truth values of the original statements and on the specific conjunctions and particles that participate in the construction of the new statement. The words and expressions “and”, “or”, “not”, “if... then”, “therefore”, “then and only then” are examples of such conjunctions. The original statements are called simple , and new statements constructed from them with the help of certain logical conjunctions - composite . Of course, the word “simple” has nothing to do with the essence or structure of the original statements, which themselves can be quite complex. In this context, the word “simple” is synonymous with the word “original”. What matters is that the truth values of simple statements are assumed to be known or given; in any case, they are not discussed in any way.
Although a statement like “Today is not Thursday” is not composed of two different simple statements, for uniformity of construction it is also considered as a compound, since its truth value is determined by the truth value of the other statement “Today is Thursday.”
Example 2. The following statements are considered as compounds:
I read Moskovsky Komsomolets and I read Kommersant.
If he said it, then it's true.
The sun is not a star.
If it is sunny and the temperature exceeds 25 0, I will arrive by train or car
Simple statements included in compounds can themselves be completely arbitrary. In particular, they themselves can be composite. The basic types of compound statements described below are defined independently of the simple statements that form them.
12. Operations on statements.
1. Negation operation.
By negating the statement A ( reads "not A", "it is not true that A"), which is true when A false and false when A– true.
Statements that deny each other A And are called opposite.
2. Conjunction operation.
Conjunction statements A And IN is called a statement denoted by A B(reads " A And IN"), the true values of which are determined if and only if both statements A And IN are true.
The conjunction of statements is called a logical product and is often denoted AB.
Let a statement be given A- “in March the air temperature is from 0 C to + 7 C" and saying IN- “It’s raining in Vitebsk.” Then A B will be as follows: “in March the air temperature is from 0 C to + 7 C and it’s raining in Vitebsk.” This conjunction will be true if there are statements A And IN true. If it turns out that the temperature was less 0 C or there was no rain in Vitebsk, then A B will be false.
3 . Disjunction operation.
Disjunction statements A And IN called a statement A B (A or IN), which is true if and only if at least one of the statements is true and false - when both statements are false.
The disjunction of statements is also called a logical sum A+B.
The statement " 4<5 or 4=5 " is true. Since the statement " 4<5 " is true, and the statement " 4=5 » – false, then A B represents the true statement " 4 5 ».
4 . Operation of implication.
By implication statements A And IN called a statement A B("If A, That IN", "from A should IN"), whose value is false if and only if A true, but IN false.
In implication A B statement A called basis, or premise, and the statement IN – consequence, or conclusion.
13. Tables of truth of statements.
A truth table is a table that establishes a correspondence between all possible sets of logical variables included in a logical function and the values of the function.
Truth tables are used for:
Calculating the truth of complex statements;
Establishing the equivalence of statements;
Definitions of tautologies.
Establishing the truth of complex statements.
Example 1. Establish the truth of a statement · C
Solution. A complex statement includes 3 simple statements: A, B, C. The columns in the table are filled with values (0, 1). All possible situations are indicated. Simple statements are separated from complex ones by a double vertical line.
When compiling a table, care must be taken not to confuse the order of actions; When filling out the columns, you should move “from the inside out,” i.e. from elementary formulas to more and more complex ones; the last column filled in contains the values of the original formula.
| A | IN | WITH | A+ | · WITH | ||
The table shows that this statement is true only in the case when A = 0, B = 1, C = 1. In all other cases it is false.
14. Equivalent formulas.
Two formulas A And IN are called equivalent if they take the same logical values for any set of values of the elementary statements included in the formula.
Equivalence is indicated by the sign " ". To transform formulas into equivalent ones, an important role is played by basic equivalences that express some logical operations through others, equivalences that express the basic laws of the algebra of logic.
For any formulas A, IN, WITH the equivalences are valid.
I. Basic equivalencies
law of idempotency
1-true
0-false
Law of contradiction
Law of the excluded middle
absorption law
splitting formulas
law of gluing
II. Equivalences expressing some logical operations through others.
de Morgan's law
III. Equivalences expressing the basic laws of logical algebra.
commutative law
association law
distributive law
15. Formulas of propositional logic.
Types of formulas of classical propositional logic– in propositional logic the following types of formulas are distinguished:
1. Laws(identically true formulas) – formulas that, under any interpretation of propositional variables, take on the value "true";
2. Controversies(identically false formulas) – formulas that, under any interpretation of propositional variables, take on the value "false";
3. Satisfiable formulas- those that take on meaning "true" for at least one set of truth values of their constituent propositional variables.
Basic laws of classical propositional logic:
1. Law of identity: ;
2. Law of contradiction:
;
3. Law of the excluded middle: ;
4. Laws of commutativity and: , ;
5. Laws of distributivity relative to , and vice versa: , ;
6. Law of removal of a true member of a conjunction: ;
7. The law of removing the false term of a disjunction: ;
8. Law of contraposition: ;
9. Laws of interexpressibility of propositional connectives: , , , , , .
Resolvability procedure- a method that allows you to determine for each formula whether it is a law, a contradiction, or a feasible formula. The most common solvability procedure is the truth table method. However, he is not the only one. An effective solvability method is the method normal forms for propositional logic formulas. Normal form A propositional logic formula is a form that does not contain the implication sign " ". There are conjunctive and disjunctive normal forms. The conjunctive form contains only conjunction signs " ". If a formula reduced to conjunctive normal form contains a subformula of the form , then the entire formula in this case is contradiction. The disjunctive form contains only the disjunction signs " ". If a formula reduced to disjunctive normal form contains a subformula of the form , then the entire formula in this case is by law. In all other cases the formula is satisfiable formula.
16. Predicates and operations on them. Quantifiers.
A sentence containing one or more variables and which, given specific values of the variables, is a statement is called expressive form or predicate.
Depending on the number of variables included in the offer, there are single, double, triple, etc. predicates, denoted accordingly: A( X), IN( X, at), WITH( X, at, z).
If a certain predicate is given, then two sets are associated with it:
1. Set (domain) of definition X, consisting of all values of variables, when substituted into a predicate, the latter turns into a statement. When specifying a predicate, its domain of definition is usually indicated.
2. Truth set T, consisting of all those values of variables, when substituting them into the predicate, a true statement is obtained.
The truth set of a predicate is always a subset of its domain of definition, that is.
You can perform the same operations on predicates as on statements.
1. Denial predicate A( X), defined on the set X, is called a predicate that is true for those values for which the predicate A( X) turns into a false statement, and vice versa.
From this definition it follows that the predicates A( X) and B( X) are not negations of each other if there is at least one value for which the predicates A( X) and B( X) turn into statements with the same truth values.
The truth set of the predicate is the complement to the truth set of the predicate A( X). Let us denote by T A the truth set of the predicate A( X), and through T - the truth set of the predicate. Then .
2. Conjunction predicates A( X) and B( XX) IN( X X X, under which both predicates turn into true statements.
The truth set of a conjunction of predicates is the intersection of the truth sets of the predicate A( X) IN( X). If we denote the truth set of the predicate A(x) by T A, and the truth set of the predicate B(x) by T B and the truth set of the predicate A(x) B(x) by , then 
3. Disjunction predicates A( X) and B( X), defined on the set X, is called the predicate A( X) IN( X), which turns into a true statement for those and only those values X X, for which at least one of the predicates turns into a true statement.
The truth set of a predicate disjunction is the union of the truth sets of the predicates forming it, i.e. .
4.By implication predicates A( X) and B( X), defined on the set X, is called the predicate A( X) IN( X), which is false for those and only those values of the variable for which the first predicate turns into a true statement, and the second into a false statement.
The truth set of the implication of predicates is the union of the truth set of the predicate B( X) with the addition to the truth set of the predicate A( X), i.e.
5. Equivalence predicates A( X) and B( X), defined on the set X, is called a predicate that turns into a true statement for all those and only those values of the variable for which both predicates turn into either true statements or false statements.
The truth set of predicate equivalence is the intersection of the truth set of a predicate with the truth set of a predicate.
Quantifier operations on predicates
A predicate can be translated into a statement using the substitution method and the “quantifier attaching” method.
About the numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 we can say: a) All these numbers are prime; b) some of the given numbers are even.
Since these sentences can be said to be true or false, the resulting sentences are statements.
If we remove the word “all” from sentence “a”, and the word “some” from sentence “b”, we get the following predicates: “the given numbers are prime”, “the given numbers are odd”.
The words “all” and “some” are called quantifiers. The word “quantifier” is of Latin origin and means “how many,” i.e. the quantifier shows how many (all or some) objects are spoken about in a particular sentence.
There are two main types of quantifiers: the general quantifier and the existence quantifier.
Terms “any”, “any”, “everyone” are called – universal quantifier. Denoted by .
Let A( X) – a certain predicate defined on the set X. Under the expression A( X) we understand the statement to be true when A( X) is true for every element of the set X, and false otherwise. R .
In example 1 for R 1 domain of definition: , set of values - . For R 2 domain of definition: , set of values: .
In many cases it is convenient to use a graphical representation of a binary relation. This is done in two ways: using points on the plane and using arrows.
In the first case, two mutually perpendicular lines are chosen as the horizontal and vertical axes. The elements of the set are plotted on the horizontal axis A and draw a vertical line through each point. The elements of the set are plotted on the vertical axis B, draw a horizontal line through each point. The intersection points of horizontal and vertical lines represent the elements of the direct product
18. Methods for specifying binary relations.
Any subset of the Cartesian product A×B is called a binary relation defined on a pair of sets A and B (in Latin, “bis” means “twice”). In the general case, by analogy with binary relations, n-ary relations can also be considered as ordered sequences of n elements taken from one of n sets.
To denote a binary relation, the sign R is used. Since R is a subset of the set A×B, we can write R⊆A×. If you need to indicate that (a, b) ∈ R, that is, there is a relation R between the elements a ∈ A and b ∈ B, then write aRb.
Methods for specifying binary relations:
1. This is the use of a rule according to which all elements included in a given relationship are indicated. Instead of a rule, you can provide a list of elements of a given relation by directly enumerating them;
2. Tabular, in the form of graphs and using sections. The basis of the tabular method is a rectangular coordinate system, where the elements of one set are plotted along one axis, and the elements of another set along the second. The intersections of the coordinates form points indicating the elements of the Cartesian product.
(Figure 1.16) shows a coordinate grid for sets. The points of intersection of three vertical lines with six horizontal lines correspond to the elements of the set A×B. The circles on the grid mark the elements of the relation aRb, where a ∈ A and b ∈ B, R denotes the “divides” relation.
Binary relations are specified by two-dimensional coordinate systems. It is obvious that all elements of the Cartesian product of three sets can similarly be represented in a three-dimensional coordinate system, four sets in a four-dimensional system, etc.;
3. The method of specifying relationships using sections is used less frequently, so we will not consider it.
19. Reflexivity of a binary relationship. Example.
In mathematics, a binary relation on a set is called reflexive if every element of this set is in a relation with itself.
The property of reflexivity for given relations by a matrix is characterized by the fact that all diagonal elements of the matrix are equal to 1; given the relationships by the graph, each element has a loop - an arc (x, x).
If this condition is not satisfied for any element of the set, then the relation is called anti-reflexive.
If the anti-reflexive relation is given by a matrix, then all diagonal elements are zero. When such a relationship is specified by a graph, each vertex does not have a loop - there are no arcs of the form (x, x).
Formally, anti-reflexivity of attitude is defined as: .
If the reflexivity condition is not satisfied for all elements of the set, the relation is said to be non-reflexive.
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1.1 . Which of the following sentences are propositions?
a) Moscow is the capital of Russia.
b) Student of the Faculty of Physics and Mathematics of the Pedagogical Institute.
c) Triangle ABC is similar to triangle A"B"C.
d) The Moon is a satellite of Mars.
f) Oxygen is a gas.
g) Porridge is a delicious dish.
h) Mathematics is an interesting subject.
i) Picasso's paintings are too abstract.
j) Iron is heavier than lead.
k) Long live the muses!
l) A triangle is called equilateral if its sides are equal.
m) If all angles in a triangle are equal, then it is equilateral.
o) The weather is bad today.
p) In the novel by A. S. Pushkin “Eugene Onegin” there are 136,245 letters.
p) The Angara River flows into Lake Baikal.
Solution. b) This sentence is not a statement because it does not state anything about the student.
c) A sentence is not a statement: we cannot determine whether it is true or false because we do not know which triangles we are talking about.
g) The sentence is not a statement, since the concept of “delicious dish” is too vague.
n) A sentence is a statement, but to find out its truth value you need to spend a lot of time.
1.2. Indicate which of the statements in the previous problem are true and which are false.
1.3. Formulate the negations of the following statements; indicate the truth values of these statements and their negations:
a) The Volga flows into the Caspian Sea.
b) The number 28 is not divisible by the number 7.
e) All prime numbers are odd.
1.4. Determine which of the statements in the following pairs are negations of each other and which are not (explain why):
a) 2< 0, 2 > 0. -
b) 6< 9, 6 9.
c) “Triangle ABC is right,” “Triangle ABC is obtuse.”
d) “Natural number n even", "Natural number n odd."
d) "Function f odd", "Function f even."
f) “All prime numbers are odd”, “All prime numbers are even.”
g) “All prime numbers are odd”, “There is a prime even number.”
h) “Man knows all the species of animals that live on Earth,” “There is a species of animal on Earth that is unknown to man.”
i) “There are irrational numbers”, “All numbers are rational”.
Solution. a) The statement “2 > 0” is not a negation of the statement “2< 0», потому что требование не быть меньше 0 оставляет две возможности: быть равным 0 и быть больше 0. Таким образом, отрицанием высказывания «2 < 0» является высказывание «2 0».
1.5. Write the following statements without the negative sign:
A) 
; V) 
;
b) 
; G) 
.
1.6.
a) Leningrad is located on the Neva and 2 + 3 = 5.
b) 7 is a prime number and 9 is a prime number.
c) 7 is a prime number or 9 is a prime number.
d) Is the number 2 even or is it a prime number?
e) 2 3, 2 3, 2 2 4, 2 2 4.
e) 2 2 = 4 or polar bears live in Africa.
g) 2 2 = 4, and 2 2 5, and 2 2 4.
Solution. a) Since both simple statements to which the conjunction operation is applied are true, therefore, based on the definition of this operation, their conjunction is a true statement.
1.7. Determine the truth values of statements A, B, C, D and E if:
- true statements, and
- false.
Solution. c) A disjunction of statements is a true statement only in the case when at least one of the constituent statements (members of the disjunction) included in the disjunction is true. In our case, the second component of the statement “2 2 = 5” is false, and the disjunction of the two statements is true. Therefore, the first component of the statement WITH true.
1.8. Formulate and write down in the form of a conjunction or disjunction the truth condition of each sentence ( A And b- real numbers):
A) 
G) 
and) 
b) 
d) 
h) 
V) 
e) 
And) 
Solution. d) A fraction is equal to zero only in the case when the numerator is equal to zero and the denominator is not equal to zero, i.e. ( A = 0) & (b 0).
1.9. Determine the truth values of the following statements:
a) If 12 is divisible by 6, then 12 is divisible by 3.
b) If 11 is divisible by 6, then 11 is divisible by 3.
c) If 15 is divisible by 6, then 15 is divisible by 3.
d) If 15 is divisible by 3, then 15 is divisible by 6.
e) If Saratov is located on the Neva, then polar bears live in Africa.
f) 12 is divisible by 6 if and only if 12 is divisible by 3.
g) 11 is divisible by 6 if and only if 11 is divisible by 3.
h) 15 is divisible by 6 if and only if 15 is divisible by 3.
i) 15 is divisible by 5 if and only if 15 is divisible by 4.
j) A body of mass m has potential energy mgh if and only if it is at its height h above the surface of the earth.
Solution. a) Since the premise statement “12 is divided by 6” is true and the consequent statement “12 is divided by 3” is true, then the compound statement based on the definition of implication is also true.
g) From the definition of equivalence we see that a statement of the form 
true if the logical meanings of the statements R And Q match, and false otherwise. In this example, both statements to which the connective “then and only then” is applied are false. Therefore the entire compound statement is true.
1.10. Let A denote the statement “9 is divisible by 3,” and let B denote the statement “8 is divisible by 3.” Determine the truth values of the following statements:
A) 
G) 
and) 
To) 
b) 
d) 
h) 
l) 
V) 
e) 
And) 
m) 
Solution. f) We have 
,
. That's why
1.11.
a) If 4 is an even number, then A.
b) If B, then 4 is an odd number.
c) If 4 is an even number, then C.
d) If D, then 4 is an odd number.
Solution. a) The implication of two statements is a false statement only in the only case when the premise is true and the conclusion is false. In this case, the premise “4 is an even number” is true and by condition the entire statement is also true. Therefore, conclusion A cannot be false, i.e. statement A is true.
1.12. Determine the truth values of statements A, B, C and D in the following sentences, of which the first two are true and the last two are false:
A) 
; b) 
;
V) 
; G) 
.
1.13. Let A denote the statement “This triangle is isosceles,” and let B denote the statement “This triangle is equilateral.” Read the following statements:
A) 
G) 
b) 
d) 
V) 
e) 
Solution. f) If a triangle is isosceles and non-equilateral, then it is not true that it is non-isosceles.
1.14. Divide the following compound statements into simple ones and write them down symbolically, introducing letter designations for their simple components:
a) If 18 is divisible by 2 and not divisible by 3, then it is not divisible by 6.
b) The product of three numbers is equal to zero if and only if one of them is equal to zero.
c) If the derivative of a function at a point is equal to zero and the second derivative of this function at the same point is negative, then this point is the maximum point of this function.
d) If in a triangle the median is not an altitude and a bisector, then this triangle is not isosceles and not equilateral.
Solution. d) Let us select and designate the simplest components of the statement as follows:
A: “In a triangle, the median is the height”;
Q: “In a triangle, the median is the bisector”;
C: “This triangle is isosceles”;
D: “This triangle is equilateral.”
Then this statement is symbolically written as follows:
1.15. From two given statements A and B, construct a compound statement using the operations of negation, conjunction and disjunction, which would be:
a) true if and only if both given statements are false;
b) false if and only if both given statements are true.
1.16. From three given statements A, B, C, construct a compound statement that is true when any one of the given statements is true, and only in this case.
1.17.
Let the statement 
true. What can be said about the logical meaning of the statement?
1.18.
If the statement 
true (false), what can be said about the logical meaning of statements:
A) 
; b) 
; V) 
; G) 
?
1.19.
If the statement 
is true and the statement 
false, what can be said about the logical meaning of the statement 
?
1.20.
Are there three such statements A, B, C such that simultaneously the statement 
was true statement 
- false and statement 
- false?
1.21. For each statement below, determine whether the information provided is sufficient to establish its logical meaning. If sufficient, then indicate this value. If this is not enough, then show that both truth values are possible:
Solution. a) Since the conclusion of the implication is true, then the entire implication will be a true statement, regardless of the logical meaning of the premises.