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

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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 property 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

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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 connected with the purposes of their study, with the context of the 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, 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 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 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.

Etymological Dictionary of the Russian Language

Identity

Greek – “the same, the same.”

Old Slavonic - tazhde (such, so).

The word is formed from a Church Slavonic pronoun according to the principle of Russian word formation and has the meaning “same, identical.”

Derivative: identical.

The beginnings of modern natural science. Thesaurus

Identity

equality (numerical, algebraic, analytical), valid at all points of the domain or for all permissible values ​​of the variables (cf. Identity).

Rhetoric: Dictionary-reference book

Identity

Identity in rhetoric: one of the top definitions, the relationship of terms of which indicates their full or partial equivalence: “Money is money”; the identity established by the top allows one to differentiate its different meanings: “Money is money, but here are rubles, and there is currency

Dictionary of linguistic terms

Identity

Correspondence of sounds, morphemes, words and phrases that have common origin. Genetic identity often does not represent a material and semantic match. Thus, the genetic identity of sounds does not mean their acoustic and articulatory coincidence. IN modern languages Genetically identical sounds may differ in their acoustic and articulatory nature. For example, [g] and [f] are genetically related sounds, although [g] is a posterior lingual stop, [g] is an anterior lingual fricative. The named sounds regularly correspond to each other in the same morphemes, differing in that after [g] there was a front vowel, and after [zh] there was a front vowel: iron (Russian), gelezis (lit.), gelsu (Prussian .);

yellow (Russian), geltas (lit.), gelb (German). Identity in rhetoric: one of the top definitions, the relationship of terms of which indicates their full or partial equivalence: “Money is money”;

The identity established by the top makes it possible to differentiate its various meanings: “Money is money, but here are rubles, and there is currency.”

Forensic Encyclopedia

Identity

(identity)

the limiting case of equality of objects, when not only all generic, but also all their individual properties coincide. In theory forensic identification the term T. denotes the presence of an object with a unique set of stable characteristics that distinguishes it from all other, including similar, objects, individualizes the object and makes it possible to recognize it in different moments time and in different states.

Philosophical Dictionary (Comte-Sponville)

Identity

Identity

♦ Identity

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, if we're talking about not about subjects, the relationship between two objects that are the same object. “In the strict sense of the word, this term is extremely precise,” Keene notes, “a thing is identical to itself and nothing else, not even a twin duplicate” (“Entities,” article “Identity”). Two monozygotic twins, even if we assume that they are exactly alike, are twins only because they are two different individuals; if they were absolutely identical (in the sense in which the author of “The Monastery of Parma” is identical with the author of “Lucien Leuven” (both novels were written by Stendhal. – Ed.)), they would constitute a single being and would not be would be twins. Thus, identity in the strict sense of the word implies uniqueness, the property of being one and the same, and no one can exactly repeat anyone other than himself.

In a broader and more traditionally rooted meaning, two objects are called identical in order to emphasize their similarity. For example, friends note the identity of points of view or tastes among themselves.

Both meanings have a right to exist, it is just important not to confuse one with the other. Therefore, when using the word “identity” in the first meaning, the definition “quantitative” is often added to it (to emphasize that we are talking about the same object: “We live in the same house”). In contrast, specific or qualitative identity indicates complete similarity between many various objects(the expression “He and I have the same car” implies the existence of two cars of the same make, same model and same color).

Identity of the latter type is never absolute (two identical cars are never absolutely the same). But can quantitative identity be absolute? In the present tense - yes, it happens, but only and exclusively in the present tense. If we consider it from the point of view of time, then it becomes as relative as qualitative identity, and perhaps even more illusory. Stendhal began writing Lucien Leuven in 1834 and was then four years younger than the author of The Cloister of Parma. What is the identity here? And if he was nevertheless identical to his later self, then why did he write a different book, and not the same one?

It would be a mistake to think that the concept of identity, formal in its essence, is capable of giving us any knowledge about reality. The assertion that Stendhal, Henri Bayle, and the author of the Life of Henri Brulard are one unit allows us to gain any knowledge only if we know what each of these words means. More precisely, just because we know this, we can claim that all three mentioned persons are one and the same person. An identity, like an identity card, communicates nothing about the content of what it points to (for it is not an essence); it only says that this content is equal to itself. A=A. Identity is not essence, although essence implies identity.

It is quite likely, in any case, I am of the opinion that nothing in time is capable of remaining identical to itself. Nothing remains permanent, as Buddhists say, and one cannot step into the same river twice. Which does not in the least prevent reality from remaining identical to itself in the present tense. At this point, Parmenides triumphs over Heraclitus, although his triumph is in vain: he wins even if Heraclitus were right. We may think that there is such a thing as identity; However, thought can only learn about what identity is through being, and not through identity itself. There is no ontology a priori. Identity is a necessary but empty concept. It is just a name that we assign to the pure presence of ourselves in reality, whereas reality is not a name.

Identity is one of the dimensions of silence that makes speech possible.

Rhetoric: Dictionary-reference book

Identity

Correspondence of sounds, morphemes, words and phrases that have a common origin. Genetic identity often does not represent a material and semantic match. Thus, the genetic identity of sounds does not mean their acoustic and articulatory coincidence. In modern languages, genetically identical sounds can differ in their acoustic and articulatory nature. For example, [g] and [zh] are genetically related sounds, although [g] is a posterior lingual stop and [zh] is an anterior fricative. The named sounds regularly correspond to each other in the same morphemes, differing in that after [g] there was a front vowel, and after [zh] there was a front vowel: iron (Russian), gelezis (lit.), gelsu (Prussian .); yellow (Russian), geltas (lit.), gelb (German).

Identity is 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 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.

Definitions, meanings of words in other dictionaries:

Philosophical Dictionary

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 in some way...

Law of identity- the principle of constancy or the principle of preservation of the subject and semantic meanings judgments (statements) in some obviously known or implied context (in conclusion, evidence, theory). It is one of the laws of classical logic.

In the process of reasoning, each concept and judgment must be used in the same sense. A prerequisite for this is the possibility of distinguishing and identifying the objects in question. . A thought about an object must have a definite, stable content, no matter how many times it is repeated. The most important property thinking - his certainty- is expressed by this logical law.

Application

In everyday life

Any acquaintance of ours changes every year, but we still distinguish him from other people we know and do not know (there is the possibility of discrimination), because he retains the main features that appear to be the same throughout the life of our acquaintance (there is the possibility of identification ). That is, in accordance with Leibniz's law(defining the concept of identity) we claim that our acquaintance has changed. However, according to law of identity we claim that this is the same person, since the definition is based on the concept of personality. The law of identity requires that we always use the same expression (name) to describe the same concept. Thus, we simultaneously consider one object (familiar) on two various levels abstractions. The possibility of distinction and identification is determined in accordance with the law of sufficient reason. IN in this case ours is used as a sufficient basis sensory perception(see identification).

In jurisprudence

In formal logic

In formal logic, the identity of a thought with itself is understood as the identity of its volume. This means that instead of a boolean variable A (\displaystyle A) into the formula " A (\displaystyle A) There is A (\displaystyle A)“thoughts of different specific content can be substituted if they have the same volume. Instead of the first A (\displaystyle A) in the formula " A (\displaystyle A) There is A (\displaystyle A)"we can substitute the concept "animal; having a soft earlobe", and instead of the second - the concept "an animal that has the ability to produce tools"(both of these thoughts are from the point of view formal logic are considered equivalent, indistinguishable, since they have the same volume, namely, the characteristics reflected in these concepts relate only to the class of people), and in this case a true judgment is obtained “An animal with a soft earlobe is an animal with the ability to produce tools.”.

In mathematics

IN mathematical logic the law of identity is the identically true implication of a logical variable with itself X ⇒ X (\displaystyle X\Rightarrow X) .

In algebra, the concept of arithmetic equality of numbers is considered as a special case general concept logical identity. However, there are mathematicians who, contrary to this point of view, do not identify the symbol " = (\displaystyle =)", found in arithmetic, with the symbol of logical identity; they don't think that equal numbers are certainly identical, and therefore consider the concept numerical equality how specific arithmetic concept. That is, they believe that the very fact of presence or absence special occasion logical identity must be determined within the framework of logic. .

Violations of the law of identity

When the law of identity is violated involuntarily, out of ignorance, then logical errors which are called