Since for any set of variable values ​​from the domain of definition of the predicate it turns into a statement, the same logical operations are defined on the set of predicates as for statements. At the same time, the content of the predicates is abstracted. Predicates are considered only in terms of their meaning. In other words, equivalent predicates do not differ.

Definition 1: By denial - local predicate
, defined on the set
, called new - local predicate defined on the same set. Indicated by:
. It reads: “it is not true that
" Predicate
evaluates to true only for those arguments for which the value of the predicate
there is a “lie” and vice versa. In other words, the predicate
satisfied by those and only those arguments that do not satisfy the given predicate
.

Double predicate
evaluates to true for those and only those variable values
from the domain of definition of the predicate, for which the predicate
takes the value “false”, i.e..

Definition 2: By conjunction - local predicate
, defined on the set
, And
- local predicate
, defined on the set
, called new
- local predicate defined on a set
, denoted by It reads: "
And
" This predicate evaluates to true only for those argument values ​​for which the predicates
And
simultaneously take the value “true”.

If, for example,
- a two-place predicate defined on a set
, A
- a unary predicate defined on a set , then the conjunction of these predicates
there is a three-place predicate defined on the set
. This new predicate evaluates to true for such triples of elements
,
,
,
, for which
And
.

Disjunction, implication and equivalence of predicates are defined similarly. The values ​​of predicates for given values ​​of free variables are determined in accordance with specific logical operations. Operations
can also be applied to predicates that have common variables. In this case, the number of variables of the resulting compound predicate will be equal to the number of different variables in its members. In particular, if operations
apply to two - local predicates depending on the same variables, then as a result of applying logical operations we get - a local predicate depending on the same variables.

Let
And
- two - local predicates depending on the same variables. Then:

a) the truth set of a conjunction is equal to the intersection of the truth sets of its members;

b) the truth set of a disjunction is equal to the union of the truth sets of its members.

It is not difficult to show that the conjunction of two predicates is identically true if and only if both given predicates are identically true. A disjunction of two predicates is satisfiable if and only if at least one of them is satisfiable. A disjunction of two predicates is identically false if and only if both given predicates are identically false. Implication of two - local predicates depending on the same arguments, is identically true if and only if its conclusion is a consequence of the premises. Equivalence of two - local predicates depending on the same variables are identically true if and only if both predicates are equivalent.

Any equation (inequality) containing variables is a predicate defined on the same set on which the equation (inequality) is given. The set of solutions to an equation (inequality) is nothing more than the truth set of the predicate. This means that when substituting the roots of an equation (or solutions to an inequality) instead of the unknowns, true statements will be obtained. If, instead of variables, numbers that are not solutions are substituted into the equation (inequality), then false statements will be obtained. Any system of equations (inequalities) can be considered as a conjunction of predicates. Solving a system means finding the domain of truth of the conjunction of predicates. A set of equations (inequalities) is nothing more than a disjunction of predicates. The equivalence of equations (inequalities) means the equivalence of the corresponding predicates.

If
, then they say that the argument
satisfies this predicate. For example, the number 3 satisfies the predicate
, and the number 1 does not satisfy him.

In mathematical logic, in addition to logical operations on predicates, there are operations quantification , which make predicate logic much richer in content compared to propositional logic. In this case, as in the case of the simplest operations, predicates are considered only from the point of view of their meanings, i.e. equivalent predicates do not differ. The main quantifier operations are: the general quantifier and the existence quantifier, which are dual to each other.

Definition 3: Let
- a unary predicate defined on a non-empty set

into a statement:
(reads: “for anyone performed
"), called general quantifier (or a universal statement). Statement
true if and only if the given predicate
identically true (i.e., the domain of truth of the predicate
coincides with the set
).

Symbol is called a general quantifier with respect to a variable , it reads: “for everyone " or "for everyone " They say that the saying
is the result of applying a general quantifier to a predicate
. Symbol comes from the English word “All” (translated: “all”).

For example, for the predicates "
" And "
", defined on the set of real numbers, the corresponding universal statements will have the form:
– “every real number is equal to itself” (true) and
– “every real number is greater than 2” (false).

Theorem 1: If
- a one-place predicate defined on a finite set consisting of
elements ,,…,, then the corresponding universal statement is equivalent to the conjunction
sayings:

Proof. Indeed, according to the definition of the general quantifier, the statement

identically true, i.e. when everything is true
statements obtained from a given predicate by replacing the variable arguments ,,…,respectively. The last remark is possible if and only if the conjunction of these
statements. Those. the terms of the equivalence are both true or false, and therefore the equivalence is proven.

The theorem shows that for predicates defined on a finite set, the operation of applying a general quantifier can be expressed through a conjunction. For predicates defined on an infinite set, this cannot be done; in this case, the operation of applying the general quantifier is completely new.

Definition 4: Let
- a unary predicate defined on a set
. Operation that transforms a predicate
into a statement
(reads: “there is , satisfying the predicate
"), called existence quantifier (or an existential statement). Statement
will be true if and only if the predicate
executable. This statement will be false if the predicate
identically false.

Symbol is called an existential quantifier with respect to a variable . It can be read: “there is such that
", or "there will be such , What
" Symbol comes from the English word "Exist" (exists).

Theorem 2: If
– a one-place predicate defined on a finite set of
elements ,,…,, then the corresponding existential statement is equivalent to the disjunction
sayings:

Proof: Definition: statement
will be false if and only if all are false
statements that are obtained from a given predicate by replacing a variable arguments ,,…,respectively. The last remark is possible if and only if the disjunction of these
statements. Those. the terms of an equivalence are both true and false, therefore the equivalence is true.

This theorem states that for predicates defined on finite sets, the operation of applying an existential quantifier can be expressed in terms of disjunction. For predicates defined on infinite sets, this cannot be done. The operation of applying the existential quantifier is then completely new.

It should be remembered that for any predicate
, defined on the set
expressions
And
These are statements, not predicates. Presence of variable here is purely external, related to the method of notation. Therefore the variable , included in the expressions
And
, called associated variable, in contrast to the variable included in the predicate
, where the variable is called free. If we apply the operation of “hanging” quantifiers to a two-place predicate
by some variable, then as a result the two-place predicate will turn into a one-place predicate with one free variable. Similar reasoning can be carried out for the second variable. The variable on which the quantifier was applied is called liaison variable. If we apply the quantifier operation to - a local predicate with respect to some variable, then it will turn into
- local predicate.

If in any predicate all the variables are related, then this predicate is a statement. For example, consider the predicate
, defined on some number set. Let's make a statement
. This is a false statement that states that there is such a number , which is greater than any number (- singular number for everyone ). By swapping the quantifiers, we get a new statement:
. This statement states that for any number you can choose a number like this , that the inequality holds
(for each there is a number ). This statement is true. It can be seen that when quantifiers are rearranged, the meaning of the statement changes. Thus, rearranging unlike quantifiers is an unacceptable operation. Quantifiers of the same name can be interchanged. Moreover, quantifiers of the same name can be combined into one, for example: . It is also unacceptable to use several quantifiers for the same variable, for example:
.

Definition 5: By a universal statement , corresponding - local predicate
, defined on the set

consistent application quantifiers of generality over variables
in any order.

This statement is designated and read briefly as follows: “for everyone
performed
».

Definition 6: By an existential statement, relevant - local predicate
, defined on the set
, is a statement obtained from
consistent application existence quantifiers over variables
in any order.

The resulting existential statement is denoted and read as follows: “there is such a set
, which is carried out
».

For example, for the two-place predicate "
» the corresponding statements have the form:
– “for any two real numbers: the first is greater than the second” (false), and
– “there are two real numbers, of which the first is greater than the second” (true).

Theorem 3: (Condition for the identical truth of a quantified predicate).

‑local predicate derived from - local predicate
, defined on the set
, by applying a general quantifier with respect to any variable, is identically true if and only if the given predicate
– identically true.

Proof: Indeed, let it be given
- local predicate
, defined on the set
. By definition, this predicate will be identically true if and only if its value for arbitrary values ​​of the arguments is “true”. This means that the universal statement is true

, defined on the set
. The last remark is possible if and only if the predicate
– identically true, but because arguments
were chosen arbitrarily, then this is equivalent to the identical truth of the given - local predicate
. The theorem has been proven.

Theorem 4: (Condition for the identical falsity of a quantified predicate).

-local predicate derived from - local predicate
, defined on the set
, by applying an existential quantifier with respect to some variable, is identically false if and only if the given predicate is identically false.

Proof: Let us have
- local predicate
, defined on the set
. It will be identically false if and only if its value for arbitrarily taken arguments
there is a "lie". This means that the existential statement is false
, corresponding to the unary predicate
, defined on the set
. The latter is possible if and only if the predicate
is identically false, and since arguments
were chosen randomly, then this - local predicate
is identically false. Q.E.D.

So far we have contrasted predicates with propositions. However, it is more convenient to count statements 0 - local predicates. Then any two true and any two false statements should be considered equivalent to each other.

The concept of a predicate

Definition 1

Predicate- a statement that contains variables that take the value $1$ or $0$ (true or false) depending on the values ​​of the variables.

Example 1

For example, the expression $x=x^5$ is a predicate because it is true for $x=0$ or $x=1$ and false for all other values ​​of $x$.

Definition 2

A set on which a predicate accepts only true values ​​is called truth set of the predicate$I_p$.

The predicate is called identically true, if on any set of arguments it evaluates to true:

$P (x_1, \dots, x_n)=1$

The predicate is called identically false, if on any set of arguments it evaluates to false:

$P (x_1, \dots, x_0)=0$

The predicate is called feasible, if it evaluates to true on at least one set of arguments.

Because predicates can take only two values ​​(true/false or $0/1$), then all operations of logical algebra can be applied to them: negation, conjunction, disjunction, etc.

Examples of predicates

Let the predicate $R(x, y)$: $“x = y”$ denote the equality relation, where $x$ and $y$ belong to the set of integers. In this case, the predicate R will be true for all equal $x$ and $y$.

Another example of a predicate is WORKS($x, y, z$) for the relation “$x$ works in city y for company $z$.”

Another example of a predicate is LIKE($x, y$) for "x likes y" for $x$ and $y$, which belong to $M$ - the set of all people.

Thus, a predicate is everything that is affirmed or denied about the subject of judgment.

Operations on predicates

Let's consider the application of logical algebra operations to predicates.

Logical operations:

Definition 3

Conjunction of two predicates $A(x)$ and $B(x)$ are a predicate that takes on a true value for those and only those values ​​of $x$ from $T$ for which each of the predicates takes on a true value and a false value at all times. in all other cases. The truth set $T$ of a predicate is the intersection of the truth sets of predicates $A(x)$ and $B(x)$. For example: predicate $A(x)$: “$x$ is an even number”, predicate $B(x)$: “$x$ is divisible by $5$.” Thus, the predicate would be "$x$ is an even number and is divisible by $5$" or "$x$ is divisible by $10$".

Definition 4

Disjunction of two predicates $A(x)$ and $B(x)$ are a predicate that evaluates to false for those and only those values ​​of $x$ from $T$ for which each of the predicates evaluates to false and evaluates to true in all others cases. The truth set of a predicate is the union of the truth domains of the predicates $A(x)$ and $B(x)$.

Definition 5

Negation of a predicate $A(x)$ is a predicate that evaluates to true for all values ​​of $x$ in $T$ for which $A(x)$ evaluates to false and vice versa. The truth set of the predicate $A(x)$ is the complement of $T"$ to the set $T$ in the set $x$.

Definition 6

Predicate implication $A(x)$ and $B(x)$ is a predicate that is false for those and only those values ​​of $x$ from $T$ for which $A(x)$ is true and $B( x)$ is false, and evaluates to true in all other cases. It reads: “If $A(x)$, then $B(x)$.”

Example 2

Let $A(x)$: “The natural number $x$ is divisible by $3$”;

$B(x)$: "The natural number $x$ is divisible by $4$."

Let’s create a predicate: “If a natural number $x$ is divisible by $3$, then it is also divisible by $4$.”

The truth set of a predicate is the union of the truth set of the predicate $B(x)$ and the complement to the truth set of the predicate $A(x)$.

In addition to logical operations, quantum operations can be performed on predicates: the use of the universal quantifier, the existence quantifier, etc.

Quantifiers

Definition 7

Quantifiers-- logical operators, the application of which to predicates turns them into false or true statements.

Definition 8

Quantifier-- logical operations that limit the domain of truth of a predicate and create a statement.

The most commonly used quantifiers are:

universal quantifier (denoted by the symbol $\forall x$) - the expression “for all $x$” (“for any $x$”);

existence quantifier (denoted by the symbol $\exists x$) - the expression “there exists $x$ such that...”;

uniqueness and existence quantifier (denoted by $\exists !x$) - the expression “there is exactly one $x$ such that...”.

In mathematical logic there is a concept tying or quantification, which denote the assignment of a quantifier to a formula.

Examples of using quantifiers

Let be the predicate “$x$ is a multiple of $7$”.

Using the universal quantifier, we can write the following false statements:

any natural number is divisible by $7$;

every natural number is divisible by $7$;

all natural numbers are divisible by $7$;

which will look like:

Picture 1.

To write true statements we use existence quantifier:

there are natural numbers that are divisible by $7$;

there is a natural number that is divisible by $7$;

at least one natural number is divisible by $7$.

The entry will look like:

Figure 2.

Let the predicate be given on the set $x$ of prime numbers: “A prime number is odd.” Putting the word “any” in front of the predicate, we get a false statement: “Any prime number is odd” (for example, $2$ is a prime even number).

We put the word “exists” in front of the predicate and get a true statement: “There is a prime number that is odd” (for example, $x=3$).

Thus, a predicate can be turned into a statement by placing a quantifier in front of the predicate.

Operations on quantifiers

To construct the negation of statements that contain quantifiers, we use rule of negation of quantifiers:

Figure 3.

Let us consider the sentences and select predicates among them, indicating the domain of truth of each of them.

1 . Negation operation.

Denial predicate P(x), given on the set X, is a predicate defined on the same set and true for those and only those values XX, under which the predicate P(x) takes on the meaning of a lie.

2 . Operation of conjunction.

Conjunction predicates P(x) And Q(x), defined on the set X, is called a predicate P(x)Q(x), given on the same set and turning into a true statement for those and only those values XX, in which both predicates take on truth values.

If we designate TR P(x), TQ- truth set of the predicate Q(x), and the truth set of their conjunction TPÙQ, then, apparently, TPÙQ = TP Ç T.Q.

Let's prove this equality.

1. Let A X and it is known that AÎ TPÙQ . By the definition of a truth set, this means that the predicate P(x)Q(x) turns into a true statement when x = a, i.e. statement R(a)Q(a) is true. Since this statement is a conjunction, then by the definition of a conjunction we obtain that each of the statements R(a) And Q(a) also true. It means that ATR And ATQ. Thus, we have shown that TPÙQ Ì TRÇ TQ.

2. Let us prove the converse statement. Let A- an arbitrary element of the set X and it is known that AÎ TP Ç TQ. By the definition of intersection of sets, this means that ATR And ATQ, from where we get that R(a) And Q(a)- true statements, therefore the conjunction of statements R(a)Q(a) will also be true. This means that the element A belongs to the truth set of the predicate P(x)Q(x), i.e. AÎ TPÙQ .

From 1 and 2, by virtue of the definition of equal sets, it follows that the equality TPÙQ =TRÇ TQ, which was what needed to be proven.

This can be visually depicted as follows.

3. Operation of disjunction.

Disjunction predicates P(x) And Q(x) is called a predicate P(x)Q(x X and turning into a true statement for those and only those values XX, for which at least one of the predicates takes on the value of truth P(x) or Q(x).

Similarly, it is proved that TPÚQ = TP È T.Q.

4 .Operation of implication.

By implication predicates P(x) And Q(x), defined on the set X, is called a predicate P(x)Q(x), defined on the same set X and turning into a false statement for those and only those values XX, at which P(x) takes on the value of truth, and Q(x)- the meaning of lies.

5 .Equivalence operation.

Equivalence predicates P(x) And Q(x), defined on the set X, is called a predicate P(x)Q(x), defined on the same set X and accepting the value of truth for those and only those values XX, for which the values ​​of each of the predicates are either true or false. The truth set in this case looks like this:

TPÛQ = .

Example. On set M=(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} predicates are given: Oh)- "number X not divisible by 5 », B(x) - « X- the number is even", C(x) - « X- the number is prime", D(x)- "number X multiple 3 " Find the truth set of the following predicates:

a) Oh)B(x); b) A(x); c) C(x)A(x); d) B(x)D(x) and depict them using Euler-Venn diagrams.

Solution: a) Find the truth set of the predicates.

A(x): T = (1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19);

B(x): T = (2, 4, 6, 8, 10, 12, 14, 16, 18, 20).

Truth set of a conjunction Oh)B(x) there are truths T And T .

Informally, a predicate can be defined as a certain statement, the meaning of which depends on the values ​​of objective variables from the set M, on which the predicate is defined.

a) P(x): “x is a prime number”;

(Here and throughout the following, to specify a predicate, we will use a short form of notation, which is described in detail as follows: “ x is a prime number.")

b) D(x,y) : “x is completely divisible by y”;

c) R(x,y): “x > y”.

Any numerical sets can be considered as a subject set for these examples, in particular, in examples a), b) – M= Í , and in c) – M= Ñ .

More strictly predicate can be defined as a mapping n th power of the set M, called the locality or arity of a predicate into a two-element set B = {1, 0}

When substituting into a predicate instead of value set subject variables we obtain a logical statement (so , a ). Thus, a predicate is a variable statement (or a system of statements), the truth of which is determined by the substitution of various values ​​of the subject variables.

Since predicates take values ​​from the set B , then logical operations ~ are defined for them. In addition, the operations of asserting universality and asserting existence are introduced for predicates.

The operation of affirmation of universality puts into correspondence the expressive form P(x) statement (read as, P(x) true for everyone x from many M, on which the predicate is defined). A statement is true if and only if the statement P(a) true for any element.

The operation assertion of existence puts into correspondence the expressive form P(x) statement (read as, there is such x from many M, for which the statement P(x) true). A statement is true if and only if the statement P(a) true for at least one element.

The signs " and $ are called quantifiers of universality and existence (quantifier translated from Latin - determination of quantity). Transition from the expressive form P(x) to statements or is called attaching a quantifier or binding a variable x(sometimes by quantification of the variable x). A variable with a quantifier is called bound, an unbound variable is called free. The meaning of bound and free variables in predicate expressions is different. A free variable is an ordinary variable that can take on different values ​​from M, and the expression P(x)– a variable statement depending on the meaning x. Expressions and do not depend on a variable x and at fixed P And M have a very definite meaning. Variables that are essentially related are not only found in mathematical logic. For example, in expressions or variable x connected, at fixed f the first expression is equal to a certain number, and the second is a function of a And b.

Thus, the statements do not speak about the properties of individual elements of the set M, but about the properties of the set itself M. The truth or falsity of these statements does not depend on how the subject variable included in them is designated, and it can be replaced by any other subject variable, for example y, and obtain statements and , having the same meaning and the same truth values ​​as the original statements.

In general, for n-ary predicate, if , the operations of asserting universality or existence can be performed k times (the order of choosing the variables for which the quantifier is assigned can be any, excluding their repetition) and get the expression

where denotes the quantifier of universality or existence. Variables in statement form (1) are bound and free.

Order relation. Ordered sets

Definition. Attitude R on a set X is called an order relation if it is transitive and asymmetric or antisymmetric.

Definition. Attitude R on a set X is called a strict order relation if it is transitive and asymmetric.

Examples relations of a strict order: “more” on the set of natural numbers, “higher” on the set of people, etc.

Definition. Attitude R on a set X is called a non-strict order relation if it is transitive and antisymmetric.

Examples relations of a non-strict order: “no more” on the set of real numbers, “be a divisor” on the set of natural numbers, etc.

Definition. A bunch of X is called ordered if an order relation is specified on it.

Example. On set X= (1; 2; 3; 4; 5) two relations are given: “ X £ at" And " X- divider at».

Both of these relations have the properties of reflexivity, antisymmetry and transitivity (construct graphs and check the properties yourself), i.e. are relations of non-strict order. But the first relation has the property of connectedness, while the second does not.

Definition. Order relation R on a set X is called a linear order relation if it has the property of connectedness.

In elementary school, many order relations are studied. Already in the first grade there are relations “less”, “more” on the set of natural numbers, “shorter”, “longer” on the set of segments, etc.

Control questions

1. Define a binary relation on a set X.

2. How to write a statement that the elements X And at are in a relationship R?

3. List ways to define relationships.

4. Formulate the properties that relationships can have. How are these properties reflected in the graph?

5. What properties must a relation have in order for it to be an equivalence relation?

6. How is the equivalence relation related to the partition of a set into classes?

7. What properties must a relation have in order for it to be a relation of order?

Chapter 5. Predicates and Theorems

In mathematics there are often sentences containing one or more variables, for example: " X+ 2 = 7”, “the city is located on the Volga”. These sentences are not statements, because it is impossible to say about them whether they are true or false. However, when substituting specific values ​​for a variable X they turn into true or false statements. So, in the first example with X= 5 we obtain a true statement, and when X= 3 – false statement.

Definition. A sentence with variables, which, given specific values ​​of the variables, turns into a statement, is called an propositional form or predicate.

Based on the number of variables included in the predicate, they are distinguished as single, double, etc. predicates and denote A(X), IN(x;y)…

Example: A(X): « X is divisible by 2" – a one-place predicate, IN(X; at): "straight X perpendicular to a straight line at" is a two-place predicate.

It should be borne in mind that the predicate may contain variables implicitly: “the number is divisible by 2”, “the student received an excellent mark on the mathematics exam.”

Specifying a predicate, as a rule, also involves specifying a set from which the values ​​of the variables included in the predicate are selected.

Definition. The set (domain) of the definition of a predicate is the set X, consisting of all values ​​of variables, when substituted into a predicate, the latter turns into a statement.

So, the predicate " X> 2" can be considered on the set of natural numbers or real numbers.

Every predicate A(X), X Î X defines a set TÌ X, consisting of elements that, when substituted into the predicate A(X) instead of X it turns out to be a true statement.

Definition. The set consisting of all those values, the substitution of which into the predicate produces a true statement, is called the truth set of the predicate (denoted T).

Example. Consider the predicate A(X): « X < 5», заданный на множестве натуральных чисел. T = {1; 2; 3; 4}.

Predicates, like statements, can be elementary or compound. Compound predicates are formed from elementary ones using logical connectives.

Let T A A(X), T V– domain of truth of the predicate IN(X).

Definition. Conjunction of predicates A(X) And IN(X) is called a predicate A(X) Ù IN(X X Î X, for which both predicates are true.

Let's show that T A Ù IN = T AÇ T V.

Proof. 1) Let A Î T A Ù IN Þ A(A) Ù IN(A) is a true statement. By definition of conjunction we have: A(A) – true, IN(A) – true Þ A Î T AÙ A Î T VÞ A Î T AÇ T VÞ T A Ù IN Ì T AÇ T V.

2) Let bÎ T AÇ T VÞ b Î T AÙ b Î T VÞ A(b) – true, IN(b) – true Þ by definition of conjunction A(b) Ù IN(b) – true statement Þ b Î T A Ù IN Þ T AÇ T VÌ T A Ù IN .

Because T A Ù IN Ì T AÇ T V And T AÇ T VÌ T A Ù IN, then by the property of equality of sets T A Ù IN = T AÇ T V, which was what needed to be proven.

Note that the resulting rule is also valid for predicates containing more than one variable.

Example. Let's look at the predicates A(X): « X < 10», IN(X): « X A(X) Ù IN(X): « X < 10 и делится на 3».

T A= {1; 2; 3; 4; 5; 6; 7; 8; 9; 10}, T V= (3; 6; 9; 12; 15; …), then T A Ù IN = {3; 6; 9}.

Definition. Predicate disjunction A(X) And IN(X) is called a predicate A(X) Ú IN(X), which is true for those and only those values X Î X, for which at least one of the predicates is true.

You can prove (on your own) that T A Ú IN = T AÈ T V.

Example. Let's look at the predicates A(X): « X divisible by 2", IN(X): « X is divisible by 3", given on the set of natural numbers. Let's find the domain of truth of the predicate A(X) Ú IN(X): « X divisible by 2 or 3."

T A= {2; 4; 6; 8; 10;…}, T V= {3; 6; 9; 12; 15; …}, T A Ú IN = {2; 3; 4; 6; 8; 9; …}.

Definition. Negation of the predicate A(X) called a predicate . It is true for those and only those values X Î X, for which the predicate A(X) is false and vice versa.

Note that = .

Definition. By implication of predicates A(X) And IN(X) is called a predicate A(X) Þ IN(X) (read: “If A(X), That IN(X)"). It turns into a false statement for those values X Î X, for which the predicate A(X) is true, and the predicate IN(X) is false.

From the definition we have that the predicate A(X) Þ IN(X) is false on the set T AÇ , and therefore is true on the complement to this set. Using the laws of operations on sets, we have: .

Control questions

1. What is called an expressive form or predicate?

2. What predicates are distinguished by the number of variables included in them? Give examples.

3. What set is called the domain of definition of the predicate?

4. What set is called the truth set of a predicate?

5. What is called a conjunction of predicates? Prove an equality connecting the domain of truth of a conjunction of predicates with the domains of truth of these predicates.

6. Give definitions of disjunction, negation, and implication of predicates. Write down equalities connecting the domains of truth of a conjunction of predicates with the domains of truth of these predicates.