Every schoolboy junior classes knows that changing the places of the terms does not change the sum; this statement is true for factors and products. That is, according to the commutative law,
a + b = b + a and
a · b = b · a.
The combinational law states:
(a + b) + c = a + (b + c) and
(ab)c = a(bc).
And the distributive law states:
a(b + c) = ab + ac.
We remembered the most elementary examples data application mathematical laws, but they all extend to very wide numerical areas.
For any value of the variable x, the meaning of the expressions 10(x + 7) and 10x + 70 are equal, since the distributive law of multiplication is satisfied for any numbers. Such expressions are said to be identically equal on the set of all numbers.
The values of the expression 5x 2 /4a and 5x/4, due to the basic property of the fraction, are equal for any value of x except 0. Such expressions are called identically equal on the set of all numbers. Except 0.
Two expressions with one variable are said to be identically equal on a set if, for any value of the variable belonging to this set, their values are equal.
Similarly, the identical equality of expressions with two, three, etc. is determined. variables on a certain set of pairs, triplets, etc. numbers.
For example, the expression 13аb and (13а)b are identically equal on the set of all pairs of numbers.
The expression 7b 2 c/b and 7bc are identically equal on the set of all pairs of values of the variables b and c in which the value of b is not equal to 0.
Equalities in which the left and right sides are expressions that are identically equal on a certain set are called identities on this set.
It is obvious that an identity on a set turns into a true numerical equality for all values of the variable (for all pairs, triplets, etc. of variable values) belonging to this set.
So, an identity is an equality with variables that is true for any values of the variables included in it.
For example, the equality 10(x + 7) = 10x + 70 is an identity on the set of all numbers; it turns into a true numerical equality for any value of x.
True numerical equalities also called identities. For example, the equality 3 2 + 4 2 = 5 2 is an identity.
In a mathematics course you have to do various transformations. For example, we can replace the sum 13x + 12x with the expression 25x. We replace the product of fractions 6a 2 /5 · 1/a with the fraction 6a/5. It turns out that the expressions 13x + 12x and 25x are identically equal on the set of all numbers, and the expressions 6a 2 /5 1/a and 6a/5 are identically equal on the set of all numbers except 0. Replacing the expression with another expression that is identically equal to it on some set, called identical transformation expressions on this set.
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What is Identity? Meaning and interpretation of the word tozhdestvo, definition of the term
1) Identity- - a relationship between objects (real or abstract), which allows us to speak of them as indistinguishable from each other, in some set of characteristics (for example, properties). In reality, all objects (things) usually differ from each other by some characteristics. This does not exclude the fact that they also have common characteristics. In the process of cognition, we identify individual things in their general characteristics, combine them into sets according to these characteristics, and form concepts about them based on the abstraction of identification (see: Abstraction). Objects that are combined into sets according to some properties they have in common cease to differ from each other, since in the process of such unification we are abstracted from their differences. In other words, they become indistinguishable, identical in these properties. If all the characteristics of two objects a and b were identical, the objects would turn into the same object. But this does not happen, because in the process of cognition we identify objects that are different from each other not by all characteristics, but only by some. Without establishing identities and differences between objects, no knowledge of the world around us, no orientation in the environment around us is possible. For the first time, in the most general and idealized formulation, the concept of the theory of two objects was given by G. W. Leibniz. Leibniz's law can be stated as follows: "x = y if and only if x has every property that y has, and y has every property that x has." In other words, an object x can be identified with an object y when absolutely all their properties are the same. The concept of T. is widely used in various sciences: in mathematics, logic and natural science. However, in all cases of its application, the identity of the objects being studied is not determined by absolutely all general characteristics, but only for some, which is related to the purposes of their study, with that context scientific theory, within which these subjects are studied.
2) Identity- a philosophical category expressing: a) equality, the sameness of an object, a phenomenon with itself, or the equality of several objects (abstract identity); b) the unity of similarity and dissimilarity, identity (in the first meaning) and difference due to change, development of the subject (specific identity). Both types of identity in the process of cognition are mutually related and transform into each other: the first of them expresses the moment of stability, the second - variability.
3) Identity- - coincidence, suggesting numeric unity.
4) Identity- - see Identity.
5) Identity- - a category expressing equality, the sameness of an object, a phenomenon with itself, or the equality of several objects. Objects A and B are said to be identical, one and the same, indistinguishable if and only if all the properties (and relations) that characterize A also characterize B, and vice versa (Leibniz’s law). However, since material reality is constantly changing, objects that are absolutely identical to themselves, even in their essential fundamentals. properties, does not happen. T. is not abstract, but concrete, i.e., containing internal differences and contradictions, constantly “removing” itself in the process of development, depending on given conditions. Identification itself individual items requires their preliminary distinction from other objects; on the other hand, one often has to identify various items(for example, for the purpose of creating their classifications). This means that T. is inextricably linked with difference and is relative. Every T. of things is temporary, transitory, but their development and change is absolute. In mathematics, where we operate with abstractions (numbers, figures) considered outside of time, outside of their measurement, Leibniz's law operates without any special restrictions. In the exact same experimental sciences the abstract, that is, abstracted from the development of things T., is used with restrictions, and only because in the process of cognition we resort, under certain conditions, to idealization and simplification of reality. The logical identity law is formulated with similar restrictions.
Identity
The relationship between objects (real or abstract), which allows us to speak of them as indistinguishable from each other, in some set of characteristics (for example, properties). In reality, all objects (things) usually differ from each other by some characteristics. This does not exclude the fact that they also have common characteristics. In the process of cognition, we identify individual things in their general characteristics, combine them into sets according to these characteristics, and form concepts about them based on the abstraction of identification (see: Abstraction). Objects that are combined into sets according to some properties they have in common cease to differ from each other, since in the process of such unification we are abstracted from their differences. In other words, they become indistinguishable, identical in these properties. If all the characteristics of two objects a and b were identical, the objects would turn into the same object. But this does not happen, because in the process of cognition we identify objects that are different from each other not by all characteristics, but only by some. Without establishing identities and differences between objects, no knowledge of the world around us, no orientation in the environment around us is possible. For the first time, in the most general and idealized formulation, the concept of the theory of two objects was given by G. W. Leibniz. Leibniz's law can be stated as follows: "x = y if and only if x has every property that y has, and y has every property that x has." In other words, an object x can be identified with an object y when absolutely all their properties are the same. The concept of T. is widely used in various sciences: mathematics, logic, and natural science. However, in all cases of its application, the identity of the objects being studied is determined not by absolutely all general characteristics, but only by some, which are related to the goals of their study, to the context of the scientific theory within which these objects are studied.
a philosophical category expressing: a) equality, the sameness of an object, a phenomenon with itself, or the equality of several objects (abstract identity); b) the unity of similarity and dissimilarity, identity (in the first meaning) and difference due to change, development of the subject (specific identity). Both types of identity in the process of cognition are mutually related and transform into each other: the first of them expresses the moment of stability, the second - variability.
Coincidence suggesting numeric unity.
See Identity.
A category expressing equality, the sameness of an object, a phenomenon with itself, or the equality of several objects. Objects A and B are said to be identical, one and the same, indistinguishable if and only if all the properties (and relations) that characterize A also characterize B, and vice versa (Leibniz’s law). However, since material reality is constantly changing, objects that are absolutely identical to themselves, even in their essential fundamentals. properties, does not happen. T. is not abstract, but concrete, i.e., containing internal differences and contradictions, constantly “removing” itself in the process of development, depending on given conditions. The very identification of individual objects requires their preliminary distinction from other objects; on the other hand, it is often necessary to identify different objects (for example, in order to create their classifications). This means that T. is inextricably linked with difference and is relative. Every T. of things is temporary, transitory, but their development and change is absolute. In mathematics, where we operate with abstractions (numbers, figures) considered outside of time, outside of their measurement, Leibniz's law operates without any special restrictions. In the exact experimental sciences, the abstract, i.e., abstract from the development of things, is used with limitations, and only because in the process of cognition we resort, under certain conditions, to idealization and simplification of reality. The logical identity law is formulated with similar restrictions.
- This the equation , which is satisfied identically, that is, valid for any admissible values of the variables included in it. From a logical point of view, Identity- This predicate , represented by the formula X = at(reads: " X identically at», « X the same as y"), which corresponds to a logical function that is true when the variables X And at mean different occurrences of the “same” object, and false in otherwise. From a philosophical (epistemological) point of view, Identity- This attitude , based on ideas or judgments about what the “same” object of reality, perception, thought is.Logical and philosophical aspects Identity additional: the first gives a formal model of the concept Identity, the second is the reasons for using this model. The first aspect includes the concept of the “same” object, but the meaning formal model does not depend on the content of this concept: the identification procedures and the dependence of the identification results on the conditions or methods of identification, on the abstractions explicitly or implicitly accepted in this case are ignored. In the second (philosophical) aspect of consideration of the basis for the use of logical models Identity are associated with how objects are identified, by what characteristics, and already depend on the point of view, on the conditions and means of identification.
Distinguishing between logical and philosophical aspects Identity goes back to the well-known position that judgment about the identity of objects and Identity as a concept, it is not the same thing (see Plato, Soch., vol. 2, M., 1970, p. 36). It is essential, however, to emphasize the independence and consistency of these aspects: the concept Identity is exhausted by the meaning of the corresponding logical function; it is not derived from the actual identity of objects, is not “extracted” from it, but is an abstraction, replenished in “suitable” conditions of experience or, in theory, through assumptions ( hypotheses ) about actually acceptable identifications; at the same time, when substitution is fulfilled (see below axiom 4) in the corresponding interval of abstraction of identification, “within” this interval, the actual Identity items exactly matches Identity in a logical sense.
Importance of the concept Identity created the need for special theories Identity The most common way to construct these theories is axiomatic. As axioms, you can specify, for example, the following (not necessarily all):
1. X = X,
2. X = at É at = X,
3. x = y & y = z É x = z,
4. A(X) É ( X = atÉ A(at)),
Where A(X) - an arbitrary predicate containing X free and free for at, A A(X) And A(at) differ only in occurrences (at least one) of variables X And y.
Axiom 1 postulates the property of reflexivity Identity In traditional logic it was considered the only logical law Identity, to which axioms 2 and 3 were usually added as “non-logical postulates” (in arithmetic, algebra, geometry). Axiom 1 can be considered epistemologically justified, since it is a kind of logical expression individuation, on which, in turn, the “givenness” of objects in experience, the possibility of their recognition, is based: in order to talk about an object “as given,” it is necessary to somehow highlight it, distinguish it from other objects and not confuse it with them in the future. In this sense Identity, based on axiom 1, is special treatment“self-identity”, which connects each object only with itself - and with no other object.
Axiom 2 postulates the property of symmetry Identity It asserts the independence of the result of identification from the order in pairs of identified objects. This axiom also has a well-known justification in experience. For example, the order of weights and goods on a scale is different when viewed from left to right for the buyer and seller facing each other, but the result is in this case the balance is the same for both.
Axioms 1 and 2 together serve abstract expression Identity as indistinguishability, a theory in which the idea of the “same” object is based on the facts of the non-observability of differences and significantly depends on the criteria of distinguishability, on the means (instruments) that distinguish one object from another, and ultimately on the abstraction of indistinguishability. Since the dependence on the “distinction threshold” is fundamentally irremovable in practice, the idea of Identity, satisfying axioms 1 and 2, is the only natural result that can be obtained in experiment.
Axiom 3 postulates transitivity Identity She states that superposition Identity there is also Identity and is the first non-trivial statement about the identity of objects. Transitivity Identity- this is either an “idealization of experience” in conditions of “decreasing accuracy”, or an abstraction that replenishes experience and “creates” a new meaning, different from indistinguishability Identity: indistinguishability is guaranteed only Identity in the interval of abstraction of indiscernibility, and this latter is not related to the fulfillment of Axiom 3. Axioms 1, 2 and 3 together serve as an abstract expression of the theory Identity How equivalence .
Axiom 4 postulates a necessary condition For Identity objects coincidence of their characteristics. From a logical point of view, this axiom is obvious: all its attributes belong to the “same” object. But since the idea of the “same” thing is inevitably based on certain kinds of assumptions or abstractions, this axiom is not trivial. It cannot be verified “in general” - according to all conceivable characteristics, but only in certain fixed intervals of abstractions of identification or indistinguishability. This is exactly how it is used in practice: objects are compared and identified not according to all conceivable characteristics, but only according to some - the main (initial) characteristics of the theory in which they want to have a concept of the “same” object based on these characteristics and on axiom 4. In these cases, the scheme of axioms 4 is replaced by a finite list of its alloforms - “meaningful” axioms congruent to it Identity For example, in axiomatic set theory Zermelo - Frenkel - axioms:
4.1 z Î x É ( x = y É z Î y),
4.2 x Î z É ( x = y É y Î z),
defining, provided that the universe contains only sets, the interval of abstraction of identification of sets by “membership in them” and by their “own membership”, with the obligatory addition of axioms 1-3, defining Identity as an equivalence.
The above axioms 1-4 refer to the so-called laws Identity From them, using the rules of logic, one can derive many other laws unknown in pre-mathematical logic. The difference between logical and epistemological (philosophical) aspects Identity does not matter as long as we are talking about general abstract formulations of laws Identity The matter, however, changes significantly when these laws are used to describe realities. Defining the concept of “one and the same” object, axiomatics Identity necessarily influence the formation of the universe “inside” the corresponding axiomatic theory.
Lit.: Tarski A., Introduction to logic and methodology of deductive sciences, trans. from English, M., 1948; Novoselov M., Identity, in the book: Philosophical Encyclopedia, t. 5, M., 1970; by him, On some concepts of the theory of relations, in the book: Cybernetics and modern scientific knowledge, M., 1976; Shreider Yu. A., Equality, similarity, order, M., 1971; Klini S.K., Mathematical logic, trans. from English, M., 1973; Frege G., Schriften zur Logik, ., 1973.
M. M. Novoselov.
Article about the word " Identity" in big Soviet Encyclopedia has been read 8308 times
This article gives a starting point idea of identities. Here we will define the identity, introduce the notation used, and, of course, give various examples identities
Page navigation.
What is identity?
It is logical to start presenting the material with identity definitions. In Makarychev Yu. N.’s textbook, algebra for 7th grade, the definition of identity is given as follows:
Definition.
Identity– this is an equality that is true for any values of the variables; any true numerical equality is also an identity.
At the same time, the author immediately stipulates that in the future this definition will be clarified. This clarification occurs in 8th grade, after becoming familiar with the definition of permissible values of variables and DL. The definition becomes:
Definition.
Identities- these are true numerical equalities, as well as equalities that are true for all acceptable values the variables included in them.
So why, when defining identity, in 7th grade we talk about any values of variables, and in 8th grade we start talking about the values of variables from their DL? Until grade 8, work is carried out exclusively with whole expressions (in particular, with monomials and polynomials), and they make sense for any values of the variables included in them. That’s why in 7th grade we say that identity is an equality that is true for any values of the variables. And in the 8th grade, expressions appear that no longer make sense not for all values of variables, but only for values from their ODZ. Therefore, we begin to call equalities that are true for all admissible values of the variables.
So identity is special case equality. That is, any identity is equality. But not every equality is an identity, but only an equality that is true for any values of the variables from their range of permissible values.
Identity sign
It is known that in writing equalities, an equal sign of the form “=” is used, to the left and to the right of which there are some numbers or expressions. If we add one more to this sign horizontal line, then it will work out identity sign“≡”, or as it is also called equal sign.
The sign of identity is usually used only when it is necessary to especially emphasize that we are faced with not just equality, but identity. In other cases, records of identities do not differ in appearance from equalities.
Examples of identities
It's time to bring examples of identities. The definition of identity given in the first paragraph will help us with this.
Numerical equalities 2=2 are examples of identities, since these equalities are true, and any true numerical equality is by definition an identity. They can be written as 2≡2 and .
Numerical equalities of the form 2+3=5 and 7−1=2·3 are also identities, since these equalities are true. That is, 2+3≡5 and 7−1≡2·3.
Let's move on to examples of identities that contain not only numbers, but also variables.
Consider the equality 3·(x+1)=3·x+3. For any value of the variable x, the written equality is true due to distributive properties multiplication relative to addition, therefore, the original equality is an example of identity. Here is another example of an identity: y·(x−1)≡(x−1)·x:x·y 2:y, here the range of permissible values of the variables x and y consists of all pairs (x, y), where x and y are any numbers except zero.
But the equalities x+1=x−1 and a+2·b=b+2·a are not identities, since there are values of the variables for which these equalities will not be true. For example, when x=2, the equality x+1=x−1 turns into the incorrect equality 2+1=2−1. Moreover, the equality x+1=x−1 is not achieved at all for any values of the variable x. And the equality a+2·b=b+2·a will turn into an incorrect equality if we take any different meanings variables a and b. For example, with a=0 and b=1 we will arrive at the incorrect equality 0+2·1=1+2·0. Equality |x|=x, where |x| - variable x is also not an identity, since it is not true for negative values x.
Examples of the most famous identities are type sin 2 α+cos 2 α=1 and a log a b =b .
In conclusion of this article, I would like to note that when studying mathematics we constantly encounter identities. Records of properties of actions with numbers are identities, for example, a+b=b+a, 1·a=a, 0·a=0 and a+(−a)=0. Also the identities are
Identity
a relationship between objects (real or abstract), which allows us to speak of them as indistinguishable from each other, in some set of characteristics (for example, properties). In reality, all objects (things) usually differ from each other by some characteristics. This does not exclude the fact that they also have common characteristics. In the process of cognition, we identify individual things in their general characteristics, combine them into sets according to these characteristics, and form concepts about them based on the abstraction of identification (see: Abstraction). Objects that are combined into sets according to some properties they have in common cease to differ from each other, since in the process of such unification we are abstracted from their differences. In other words, they become indistinguishable, identical in these properties. If all the characteristics of two objects a and b were identical, the objects would turn into the same object. But this does not happen, because in the process of cognition we identify objects that are different from each other not by all characteristics, but only by some. Without establishing identities and differences between objects, no knowledge of the world around us, no orientation in the environment around us is possible.
For the first time, in the most general and idealized formulation, the concept of the theory of two objects was given by G. W. Leibniz. Leibniz's law can be stated as follows: "x = y if and only if x has every property that y has, and y has every property that x has." In other words, an object x can be identified with an object y when absolutely all their properties are the same. The concept of T. is widely used in various sciences: mathematics, logic, and natural science. However, in all cases
its application, the identity of the objects being studied is determined not by absolutely all general characteristics, but only by some, which are related to the goals of their study, to the context of the scientific theory within which these objects are studied.
Dictionary of logic. - M.: Tumanit, ed. VLADOS center. A.A.Ivin, A.L.Nikiforov. 1997 .
Synonyms:See what “identity” is in other dictionaries:
Identity- Identity ♦ Identité Coincidence, the property of being the same. Same as what? The same as the same, otherwise it will no longer be identity. Thus, identity is, first of all, the relation of oneself to oneself (my identity is myself) or... Philosophical Dictionary Sponville
A concept that expresses the limiting case of equality of objects, when not only all generic, but also all their individual properties coincide. The coincidence of generic properties (similarity), generally speaking, does not limit the number of equated... ... Philosophical Encyclopedia
Cm … Synonym dictionary
The relationship between objects (objects of reality, perception, thought) considered as one and the same; limiting case of equality relation. In mathematics, an identity is an equation that is satisfied identically, that is, valid for... ... Big Encyclopedic Dictionary
IDENTITY, a and IDENTITY, a, cf. 1. Complete similarity, coincidence. T. views. 2. (identity). In mathematics: equality that is valid for any numerical values the quantities included in it. | adj. identical, aya, oe and identical, aya, oh (to 1... ... Ozhegov's Explanatory Dictionary
identity- IDENTITY is a concept usually represented in natural language either in the form “I (am) the same as b, or “a is identical to b,” which can be symbolized as “a = b” (such a statement is usually called an absolute T.), or in the form ... ... Encyclopedia of Epistemology and Philosophy of Science
identity- (incorrect identity) and outdated identity (preserved in the speech of mathematicians, physicists) ... Dictionary of difficulties of pronunciation and stress in modern Russian language
AND DISTINCTION are two interrelated categories of philosophy and logic. When defining the concepts of T. and R., two fundamental principles are used: the principle of individuation and the principle of T. indistinguishable. According to the principle of individuation, which has been meaningfully developed... History of Philosophy: Encyclopedia
English identity; German Identity. 1. In mathematics, an equation that is valid for all valid values of the arguments. 2. The limiting case of equality of objects, when not only all generic, but also all their individual properties coincide. Antinazi.… … Encyclopedia of Sociology
- (designation ≡) (identity, symbol ≡) An equation that is true for any values of the variables included in it. Thus, z ≡ x + y means that z is always the sum of x and y. Many economists are sometimes inconsistent and use the usual sign even then... Economic dictionary
identity- identity, personal identification ID - [] Topics information protection Synonyms identity, personal identificationID EN identityID ... Technical Translator's Guide
Books
- Difference and identity in Greek and medieval ontology, R. A. Loshakov. The monograph examines the main issues of Greek (Aristotelian) and medieval ontology in the light of the understanding of being as Difference. This demonstrates the derivative, secondary,...