Explanatory dictionary of the Russian language. S.I. Ozhegov, N.Yu. Shvedova.

identity

A and IDENTITY. -a, cf.

    Full similarity, coincidence. G. views.

    (identity). In mathematics: an equality that is valid for any numerical values ​​of its constituent quantities. || adj. identical, -th, -th and identical, -th, -th (to 1 value). Identity algebraic expressions. ALSO [do not mix with a combination of the pronoun "that" and the particle "same"].

    1. adv. In the same way, just like anyone else. You are tired, I

      union. Same as also. Are you leaving, brother? - T.

    particle. Expresses distrustful or negative, ironic attitude (simple). *T. smart guy found! He is a poet. - Poet comrade (to me)!

New explanatory and derivational dictionary of the Russian language, T. F. Efremova.

identity

    1. Absolute coincidence with smth., smth. both in its essence and in external signs and manifestations.

      An exact match. something

    2. cf. An equality that is valid for all the numerical values ​​​​of the letters included in it (in mathematics).

Encyclopedic Dictionary, 1998

identity

the relationship between objects (objects of reality, perception, thought) considered as "one and the same"; "limiting" case of the relation of equality. In mathematics, an identity is an equation that is satisfied identically, i.e. is valid for any admissible values ​​of the variables included in it.

Identity

the basic concept of logic, philosophy and mathematics; used in the languages ​​of scientific theories to formulate defining relations, laws and theorems. In mathematics, T. ≈ is an equation that is satisfied identically, that is, it is valid for any admissible values ​​of the variables included in it. From a logical point of view, T. ≈ is a predicate represented by the formula x \u003d y (read: "x is identical to y", "x is the same as y"), which corresponds to a logical function that is true when the variables x and y mean different occurrences of the "same" item, and false otherwise. From a philosophical (epistemological) point of view, T. is an attitude based on ideas or judgments about what the “one and the same” object of reality, perception, thought is. The logical and philosophical aspects of T. are additional: the first gives a formal model of the concept of T., the second - the basis for the application of this model. The first aspect includes the concept of “one and the same” subject, but the meaning of the formal model does not depend on the content of this concept: the procedures of identifications and the dependence of the results of identifications on the conditions or methods of identifications, on the abstractions explicitly or implicitly accepted in this case are ignored. In the second (philosophical) aspect of consideration, the grounds for applying the logical models of T. are associated with how objects are identified, by what signs, and already depend on the point of view, on the conditions and means of identification. The distinction between the logical and philosophical aspects of T. goes back to the well-known position that the judgment of the identity of objects and T. as a concept is not the same thing (see Platon, Soch., vol. 2, M., 1970, p. 36) . It is essential, however, to emphasize the independence and consistency of these aspects: the concept of logic is exhausted by the meaning of the logical function corresponding to it; it is not deduced from the actual identity of objects, “is not extracted” from it, but is an abstraction replenished under “suitable” conditions of experience or, in theory, by assumptions (hypotheses) about actually admissible identifications; at the same time, when substitution (see axiom 4 below) is fulfilled in the corresponding interval of the abstraction of identification, "inside" this interval, the actual T. of objects coincides exactly with T. in the logical sense. The importance of the concept of T. has led to the need for special theories of T. The most common way of constructing these theories is axiomatic. As axioms, you can specify, for example, the following (not necessarily all):

    x = y É y = x,

    x = y & y = z É x = z,

    A (x) É (x = y É A (y)),

    where A (x) ≈ an arbitrary predicate containing x freely and free for y, and A (x) and A (y) differ only in the occurrences (at least one) of the variables x and y.

    Axiom 1 postulates the property of reflexivity of T. In traditional logic, it was considered the only logical law of T., to which, usually (in arithmetic, algebra, geometry), axioms 2 and Z were added as “non-logical postulates”. Axiom 1 can be considered epistemologically justified, since it is a kind of logical expression of individuation, on which, in turn, the “givenness” of objects in experience, the possibility of recognizing them, is based: in order to speak of an object “as given”, it is necessary to somehow single it out, distinguish it from other objects, and in the future not to be confused with them. In this sense, T., based on Axiom 1, is a special relation of "self-identity" that connects each object only with itself ≈ and with no other object.

    Axiom 2 postulates the symmetry property T. It asserts the independence of the result of identification from the order in pairs of identified objects. This axiom also has a certain justification in experience. For example, the order of the weights and goods on the balance is different, from left to right, for the buyer and seller facing each other, but the result - in this case, the equilibrium - is the same for both.

    Axioms 1 and 2 together serve as an abstract expression of T. as indistinguishability, a theory in which the idea of ​​the “same” object is based on the facts of the non-observability of differences and essentially depends on the criteria of distinguishability, on the means (devices) that distinguish one object from another , ultimately ≈ from the abstraction of indistinguishability. Since the dependence on the “threshold of distinguishability” cannot be eliminated in principle in practice, the idea of ​​a temperature that satisfies axioms 1 and 2 is the only natural result that can be obtained experimentally.

    Axiom 3 postulates the transitivity of T. It states that the superposition of T. is also T. and is the first non-trivial statement about the identity of objects. The transitivity of T. is either an “idealization of experience” under conditions of “decreasing accuracy,” or an abstraction that replenishes experience and “creates” a new, different from indistinguishability, meaning of T.: indistinguishability guarantees only T. in the interval of abstraction of indistinguishability, and this latter does not connected with the fulfillment of Axiom 3. Axioms 1, 2, and 3 together serve as an abstract expression of the theory of T. as an equivalence.

    Axiom 4 postulates that a necessary condition for the typology of objects is the coincidence of their characteristics. From a logical point of view, this axiom is obvious: “one and the same” object has all its features. But since the notion 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 signs, 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 signs, but only according to some - the main (initial) signs of the theory in which they want to have a concept of the "same" object based on these signs and on axiom 4. In these cases, the scheme of axioms 4 is replaced by a finite list of its alloforms ≈ "meaningful" axioms T congruent to it. For example, in the axiomatic set theory of Zermelo ≈ Frenkel ≈ axioms:

    4.1 z О x О (x = y О z О 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 the identification of sets according to their “membership in them” and according to their “own membership”, with the obligatory addition of axioms 1≈3, defining T. as equivalence.

    The axioms 1≈4 listed above refer to the so-called laws of T. From them, using the rules of logic, one can derive many other laws that are unknown in pre-mathematical logic. The distinction between the logical and epistemological (philosophical) aspects of theory is irrelevant as long as we are talking about general abstract formulations of the laws of theory. The matter, however, changes significantly when these laws are used to describe realities. Defining the concept of “one and the same” object, the axiomatics of theory necessarily influence the formation of the universe “within” the corresponding axiomatic theory.

    Lit .: Tarsky A., Introduction to the logic and methodology of deductive sciences, trans. from English, M., 1948; Novoselov M., Identity, in the book: Philosophical Encyclopedia, v. 5, M., 1970; his, On some concepts of the theory of relations, in the book: Cybernetics and modern scientific knowledge, M., 1976; Shreyder Yu. A., Equality, similarity, order, M., 1971; Klini S. K., Mathematical logic, trans. from English, M., 1973; Frege G., Schriften zur Logik, B., 1973.

    M. M. Novoselov.

Wikipedia

Identity (mathematics)

Identity(in mathematics) - equality, which is satisfied on the entire set of values ​​​​of the variables included in it, for example:

a − b = (a + b)(a − b) (a + b) = a + 2ab + b

etc. Sometimes an identity is also called an equality that does not contain any variables; e.g. 25 = 625.

Identical equality, when they want to emphasize it especially, is indicated by the symbol " ≡ ".

Identity

Identity, identity- polysemantic terms.

  • An identity is an equality that holds on the entire set of values ​​of its constituent variables.
  • Identity is a complete coincidence of the properties of objects.
  • Identity in physics is a characteristic of objects, in which the replacement of one of the objects with another does not change the state of the system while maintaining these conditions.
  • The law of identity is one of the laws of logic.
  • The principle of identity is the principle of quantum mechanics, according to which the states of a system of particles, obtained from each other by rearranging identical particles in places, cannot be distinguished in any experiment, and such states should be considered as one physical state.
  • "Identity and Reality" - a book by E. Meyerson.

Identity (philosophy)

Identity- a philosophical category expressing equality, the sameness of an object, phenomenon with itself or the equality of several objects. Objects A and B are said to be identical, the same, if and only if all properties. This means that identity is inextricably linked with difference and is relative. Any identity of things is temporary, transient, while their development, change is absolute. In the exact sciences, however, abstract identity, i.e., abstracted from the development of things, in accordance with Leibniz's law, is used because in the process of cognition, idealization and simplification of reality are possible and necessary under certain conditions. The logical law of identity is also formulated with similar restrictions.

Identity should be distinguished from similarity, similarity and unity.

Similar we call objects that have one or more common properties; the more objects have common properties, the closer their similarity comes to identity. Two objects are considered identical if their qualities are exactly the same.

However, it should be remembered that in the objective world there can be no identity, since two objects, no matter how similar they are in quality, still differ in number and the space they occupy; only where material nature rises to spirituality does the possibility of identity appear.

The necessary condition for identity is unity: where there is no unity, there can be no identity. The material world, divisible to infinity, does not possess unity; unity comes with life, especially with spiritual life. We speak of the identity of an organism in the sense that its one life persists despite the constant change of particles that make up the organism; where there is life, there is unity, but in the true meaning of the word there is still no identity, since life waxes and wanes, remaining unchanged only in the idea.

The same can be said about personalities- the highest manifestation of life and consciousness; and in personality we only assume identity, but in reality there is none, since the very content of personality is constantly changing. True identity is possible only in thinking; a properly formed concept has an eternal value regardless of the conditions of time and space in which it is conceived.

Leibniz, with his principium indiscernibilium, established the idea that two things cannot exist that are completely similar in qualitative and quantitative respects, since such similarity would be nothing but identity.

The philosophy of identity is the central idea in the works of Friedrich Schelling.

Examples of the use of the word identity in the literature.

This is precisely the great psychological merit of both ancient and medieval nominalism, that it thoroughly dissolved the primitive magical or mystical identity words with an object are too thorough even for a type whose foundation is not to cling tightly to things, but to abstract the idea and place it above things.

This identity subjectivity and objectivity, and constitutes precisely the universality now attained by self-consciousness, which rises above the two sides or particularities mentioned above and dissolves them in itself.

At this stage, self-conscious subjects correlated with each other have risen, therefore, through the removal of their unequal singularity of individuality, to the consciousness of their real universality - their inherent freedom - and thereby to the contemplation of a certain identities them with each other.

A century and a half later, Inta, the great-great-great-granddaughter of the woman who was given a seat in the spaceship by Sarp, amazed by her inexplicable identity with Vella.

But when it turned out that before his death, the good writer Kamanin read the manuscript of KRASNOGOROV, and at the same time the very one whose candidacy was discussed by the ferocious physicist Sherstnev a second before his, Sherstnev’s, SIMILAR death, - then, you know, it smelled on me of not simple coincidence, it smells IDENTITY!

The merit of Klossowski is that he showed that these three forms are now connected forever, but not due to dialectical transformation and identity opposites, but through their dispersion over the surface of things.

In these works, Klossowski develops the theory of the sign, meaning and nonsense, and also gives a deeply original interpretation of Nietzsche's idea of ​​​​the eternal return, understood as an eccentric ability to assert divergences and disjunctions, leaving no room for identity me, neither identity peace or identity God.

As in any other type of identification of a person by appearance, in a photo-portrait examination, the identified object in all cases is a specific individual, identity which is being installed.

Now a teacher has emerged from the student, and above all, as a teacher, he coped with the great task of the first period of his master's degree, having won the struggle for authority and full identity person and position.

But in the early classics it identity thinking and conceivable was interpreted only intuitively and only descriptively.

For Schelling identity Nature and Spirit is a natural-philosophical principle that precedes empirical knowledge and determines the understanding of the results of the latter.

Based on this identities mineral features and it is concluded that this Scottish formation is contemporary with the lowest formations of Wallis, because the amount of available paleontological data is too small to confirm or refute this kind of position.

Now it is no longer the origin that gives place to historicity, but the very fabric of historicity reveals the need for the origin, which would be both internal and external, like some hypothetical apex of a cone, where all differences, all scattering, all discontinuities are compressed into a single point. identities, into that incorporeal image of the Identical, capable, however, of splitting and turning into the Other.

It is known that there are often cases when an object to be identified from memory does not have a sufficient number of noticeable features that would allow it to be identified. identity.

It is clear, therefore, that veche, or uprisings, in Moscow against people who wanted to flee from the Tatars, in Rostov against the Tatars, in Kostroma, Nizhny, Torzhok against the boyars, veches convened by all the bells, should not, one by one. identity names, mixed with the vechas of Novgorod and other old cities: Smolensk, Kiev, Polotsk, Rostov, where the inhabitants, according to the chronicler, converged as if on a thought, for a vecha, and that the elders decided, the suburbs agreed to that.

LECTURE №3 Proof of identities

Purpose: 1. Repeat the definitions of identity and identically equal expressions.

2.Introduce the concept of identical transformation of expressions.

3. Multiplication of a polynomial by a polynomial.

4. Decomposition of a polynomial into factors by the grouping method.

May every day and every hour

We will get something new

Let our minds be good

And the heart will be smart!

There are many concepts in mathematics. One of them is identity.

An identity is an equality that holds for all values ​​of the variables that are included in it. We already know some of the identities.

For example, all abbreviated multiplication formulas are identities.

Abbreviated multiplication formulas

1. (a ± b)2 = a 2 ± 2 ab + b 2,

2. (a ± b)3 = a 3 ± 3 a 2b + 3ab 2 ± b 3,

3. a 2 - b 2 = (a - b)(a + b),

4. a 3 ± b 3 = (a ± b)(a 2 ab + b 2).

Prove Identity- this means to establish that for any admissible value of the variables, its left side is equal to the right side.

There are several different ways of proving identities in algebra.

Ways to prove identities

    Perform equivalent transformations left side of the identity. If in the end we get the right side, then the identity is considered proven. Perform equivalent transformations the right side of the identity. If in the end we get the left side, then the identity is considered proven. Perform equivalent transformations left and right sides of the identity. If we get the same result as a result, then the identity is considered proven. Subtract the left side from the right side of the identity. We perform equivalent transformations on the difference. And if in the end we get zero, then the identity is considered proven. Subtract the right side from the left side of the identity. We perform equivalent transformations on the difference. And if in the end we get zero, then the identity is considered proven.

It should also be remembered that the identity is valid only for admissible values ​​of variables.

As you can see, there are many ways. Which way to choose in this particular case depends on the identity you need to prove. As you prove various identities, experience will come in choosing the method of proof.

An identity is an equation that is satisfied identically, that is, it is valid for any admissible values ​​of its constituent variables. To prove an identity means to establish that for all admissible values ​​of the variables, its left and right sides are equal.
Ways to prove identity:
1. Transform the left side and get the right side as a result.
2. Perform transformations on the right side and finally get the left side.
3. Separately, the right and left parts are transformed and the same expression is obtained in the first and second cases.
4. Compose the difference between the left and right parts and, as a result of its transformations, get zero.
Let's look at a few simple examples

Example 1 Prove Identity x (a + b) + a (b-x) = b (a + x).

Solution.

Since there is a small expression on the right side, let's try to transform the left side of the equality.

x (a + b) + a (b-x) = x a + x b + a b - a x.

We present like terms and take the common factor out of the bracket.

x a + x b + a b – a x = x b + a b = b (a + x).

We got that the left side after the transformations became the same as the right side. Therefore, this equality is an identity.

Example 2 Prove the identity: a² + 7a + 10 = (a+5)(a+2).

Solution:

In this example, you can do the following. Let's open the brackets on the right side of the equality.

(a+5) (a+2) = (a²) + 5 a +2 a +10 = a² + 7 a + 10.

We see that after the transformations, the right side of the equality has become the same as the left side of the equality. Therefore, this equality is an identity.

"The replacement of one expression by another identically equal to it is called the identical transformation of the expression"

Find out which equality is an identity:

1. - (a - c) \u003d - a - c;

2. 2 (x + 4) = 2x - 4;

3. (x - 5) (-3) \u003d - 3x + 15.

4. pxy (- p2 x2 y) = - p3 x3 y3.

“To prove that some equality is an identity, or, as they say, to prove an identity, one uses identical transformations of expressions”

The equality is true for any values ​​of the variables, called identity. To prove that some equality is an identity, or, as they say otherwise, to prove identity, use identical transformations of expressions.
Let's prove the identity:
xy - 3y - 5x + 16 = (x - 3)(y - 5) + 1
xy - 3y - 5x + 16 = (xy - 3y) + (- 5x + 15) +1 = y(x - 3) - 5(x -3) +1 = (y - 5)(x - 3) + 1 As a result identity transformation left side of the polynomial, we obtained its right side and thus proved that this equality is identity.
For identity proofs transform its left-hand side into a right-hand side or its right-hand side into a left-hand side, or show that the left and right sides of the original equality are identically equal to the same expression.

Multiplication of a polynomial by a polynomial

Let's multiply the polynomial a+b to a polynomial c + d. We compose the product of these polynomials:
(a+b)(c+d).
Denote the binomial a+b letter x and transform the resulting product according to the rule of multiplication of a monomial by a polynomial:
(a+b)(c+d) = x(c+d) = xc + xd.
In expression xc + xd. substitute instead of x polynomial a+b and again use the rule for multiplying a monomial by a polynomial:
xc + xd = (a+b)c + (a+b)d = ac + bc + ad + bd.
So: (a+b)(c+d) = ac + bc + ad + bd.
Product of polynomials a+b And c + d we have presented in the form of a polynomial ac+bc+ad+bd. This polynomial is the sum of all monomials obtained by multiplying each term of the polynomial a+b for each member of the polynomial c + d.
Conclusion: the product of any two polynomials can be represented as a polynomial.
rule: to multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other polynomial and add the resulting products.
Note that when multiplying a polynomial containing m terms on a polynomial containing n members in the product, before reduction of similar members, it should turn out mn members. This can be used for control.

Decomposition of a polynomial into factors by the grouping method:

Earlier, we got acquainted with the decomposition of a polynomial into factors by taking the common factor out of brackets. Sometimes it is possible to factorize a polynomial using another method - grouping of its members.
Factoring the polynomial
ab - 2b + 3a - 6
ab - 2b + 3a - 6 = (ab - 2b) + (3a - 6) = b(a - 2) + 3(a - 2) Each term in the resulting expression has a common factor (a - 2). Let's take this common factor out of brackets:
b(a - 2) + 3(a - 2) = (b + 3)(a - 2) As a result, we factored the original polynomial:
ab - 2b + 3a - 6 = (b + 3)(a - 2) The method we used to factorize a polynomial is called way of grouping.
Polynomial decomposition ab - 2b + 3a - 6 can be multiplied by grouping its terms differently:
ab - 2b + 3a - 6 = (ab + 3a) + (- 2b - 6) = a(b + 3) -2(b + 3) = (a - 2)(b + 3)

Repeat:

1. Ways of proving identities.

2. What is called the identical transformation of an expression.

3. Multiplication of a polynomial by a polynomial.

4. Factorization of a polynomial by the grouping method

Identity in mathematics is a very commonly used concept. There are concepts of identical equalities, identical expressions and identical transformations, let's take a closer look at what each of these concepts means.

Identity expressions in mathematics

Consider three simple algebraic expressions:

  • $5x + 10$;
  • $(x + 2) \cdot 5$
  • $\frac(20x + 40)(4)$

Regardless of the values ​​of $x$ used, all three expressions are equal to each other.

In order to prove this, we use elementary transformations that are allowed in mathematics, and we get that $5x + 10 = 5x + 10 = 5x + 10$, that is, all three expressions are equal to each other. Simplifying, it becomes clear that no matter what $x$ is chosen, these expressions will always be equal.

We come directly to the definition of identical expressions:

Definition 1

Expressions are called identical with each other if, for any values ​​of the variables, they are always equal to each other.

For example, the expression $5x + 10$ can be said to be identical to the expressions $(x + 2) \cdot 5$ and $\frac(20x + 40)(4)$.

It is also worth paying attention to the fact that expressions are not always identical for all possible values ​​of variables, for example, the expressions $\frac(y^2-4)(y-2)$ and $y+2$ are identical for any $y$, except for $y=2$.

When the value of y is equal to two, the first of these two expressions loses its meaning, since it is impossible to divide by zero, and zero is obtained in the denominator at this value.

These expressions can be called identical for all admissible values ​​of the variable $y$, that is, these expressions are identical for all $y$, for which both expressions do not lose their meaning. Such expressions are called identical on a given set of values.

The concepts of "identity" and "identical equality"

What is an identity in algebra?

Definition 2

An identity in mathematics is an equality that always holds or, in other words, is valid for all sets of values ​​of its variables.

If two or more identical expressions are written directly next to each other through the “equal” sign, then we get identical equality, that is, identity.

The same equalities include the commutative law of addition $a+b =b + a$ and the associative law of multiplication $(ab) \cdot c = a \cdot (bc)$, since they are true regardless of the value of the variables $a, b ,c$. Shortcut formulas for difference of squares, squares of difference, and squares of sum are other examples of identical equalities.

Sometimes not only expressions containing some variables are called identities, but also all arithmetically true equalities of the type $2+2=4$.

Not any equality containing variables can be called an identity. For example, the equality $y+5 = 7$ is observed only for $y= 2$, for any other value of $y$ it is not observed and therefore it cannot be called an identity.

Identity sign in mathematics

Definition 3

Most often, identities are written through the “equal” sign - “$ = $”, the sign “identically” - “≡” is sometimes used to highlight the identity of any equality in speech. Usually, the identity sign is used much less frequently than the equal sign.

Identity transformations

Very often, in order to simplify the process of calculating any expressions, as well as to compare them and more convenient substitution of variables into equalities, various mathematical transformations are used. These transformations are called identical transformations, since they do not change the final values ​​of expressions and equalities.

Definition 4

Identical transformations are transformations and replacements of one expression by another, identical to it, which do not change the final value of the expressions and do not lead to a violation of the identity of equalities.

Any expression, for any valid values ​​of the variables used in it, takes some value. From this we can conclude that the application of various laws observed for arithmetic operations leads to the transformation of the original expression into a new one, identical to the original expression.

Example 1

What expressions are identical?

  1. $(10 + 3)$ and $13 \cdot (1 +5)$.
  2. $(x^2 + y^2)$ and $(x – y)(x+y)$.
  3. $8$ and $(2 \cdot 3 + 16 – 14)$.
  4. $7 + 4$ and $6 + 6$.

Answer:

Expressions numbered 2 and 3 are identical, in the case of expressions numbered 2, the abbreviated formula for the difference of squares is given on the left, and the expanded formula is given on the right. In the case of the third expression, you need to simplify the expression on the right:

$(2 \cdot 3 + 16 - 14)= 6 + 16 - 14 = 8$

Both parts of which are identically equal expressions. Identities are divided into letter and number.

Identity expressions

The two algebraic expressions are called identical(or identically equal), if for any numerical values ​​of the letters they have the same numerical value. These are, for example, the expressions:

x(5 + x) and 5 x + x 2

Both presented expressions, for any value x will be equal to each other, so they can be called identical or identically equal.

Numeric expressions that are equal to each other can also be called identical. For example:

20 - 8 and 10 + 2

Letter and number identities

Letter identity is an equality that is valid for any values ​​of the letters included in it. In other words, such an equality, in which both parts are identically equal expressions, for example:

(a + b)m = am + bm
(a + b) 2 = a 2 + 2ab + b 2

Numeric Identity- this is an equality containing only numbers expressed in digits, in which both parts have the same numerical value. For example:

4 + 5 + 2 = 3 + 8
5 (4 + 6) = 50

Identity transformations of expressions

All algebraic operations are the transformation of one algebraic expression into another, identical to the first.

When calculating the value of an expression, opening brackets, taking the common factor out of brackets, and in a number of other cases, some expressions are replaced by others that are identically equal to them. The replacement of one expression by another, identically equal to it, is called identical transformation of the expression or simply expression conversion. All expression conversions are performed based on the properties of operations on numbers.

Consider the identical transformation of the expression using the example of taking the common factor out of brackets:

10x - 7x + 3x = (10 - 7 + 3)x = 6x