Identity equation

≡

Identity mark

An identity equation , often called identity for short , is a mathematical statement written as an equation about the equality of expressions, formulas or functions in certain areas of definition.

Explanation

Identity equations contain variables . However, it is not a question of determining this, rather it is claimed that both sides of the equation, which result from substituting any elements of the agreed definition range in place of the variables, lead to the same value.

As an example, the binomial formula

${\displaystyle (a+b)^{2}\equiv a^{2}+2ab+b^{2}}$   for all ${\displaystyle a,b\in \mathbb {R} }$

considered. This identity means that no matter which real numbers are used for or , the value of the left side, the square of the sum of the numbers used for and , equals the value of the right side, the sum of the individual squares plus twice the product from the numbers used for and , is. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$

The domain of definition used here is common because it often corresponds to the level of knowledge of the user of this formula. If one imagines a school level at which the pupil has only become acquainted with the rational numbers , i.e. the fractions, but not yet the real numbers, then one will state the above identity with the smaller domain instead of . When you finally got to know the complex numbers , you will use the larger domain . ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {Q} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \mathbb {C} }$

In more advanced mathematics you get to know domains of definition in which this binomial formula no longer applies, e.g. B. the set of 2 × 2 matrices . This shows that the existence of identities depends on the chosen domain. This must therefore always be agreed, this is done either tacitly or through explicit information.

Another very well known identity is

${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$   for everyone . ${\displaystyle x\in \mathbb {R} }$

Again, it's not about determining the value of the variable x . This is not possible because the theorem holds for every conceivable value of x . Using the x twice simply stipulates that the same value should be used both times. This identity says: If you determine the sine and cosine values of any real number , squar them and then add them, you always get the result 1.

For clarification, use - especially with identities for functions of real numbers or other domains of definition - instead of the equal sign the symbol "≡" and reads "is identical", for example:

${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)\equiv 1}$   for everyone . ${\displaystyle x\in \mathbb {R} }$

This notation is a convention that is particularly widespread in technical disciplines; it does not change the meaning presented above. This symbol is particularly popular there for the equality of functions. However, equality is still meant, so that the use of the other character “≡” can also cause confusion, especially since this character is used by many authors for the modulo operation. The use of these symbols is usually specified in the introductory sections of textbooks, so that in case of doubt these introductory sections should be consulted.

Formal definition

Formally, an identity is a universal quantification, with the predicate being an equality relation. There is probably nothing to prevent more general equivalence relations from being allowed as a predicate.

Let " " be the equality relation on a set . Every statement of form is an identity ${\displaystyle =}$ ${\displaystyle M}$

${\displaystyle \forall x\in G\colon \,T_{1}(x)=T_{2}(x),}$

where functions are off with . The definition set is also called the “basic set” in the context of the universal quantifier. ${\displaystyle T_{1},T_{2}}$ ${\displaystyle \mathrm {Abb} (G,Z)}$ ${\displaystyle Z\subseteq M}$ ${\displaystyle G}$

It should be noted that z. B. can also be a tuple . Independent of this, z. B. also have tuples as a value. Therefore, equations with several variables and systems of equations are included in the definition. ${\displaystyle x}$ ${\displaystyle T_{1},T_{2}}$

Example: With and the following applies: ${\displaystyle x:=(x_{1},x_{2})}$ ${\displaystyle G:=\mathbb {R} \times \mathbb {R} }$

${\displaystyle \forall x\in G\colon \,\sin(x_{1}+x_{2})=\sin(x_{1})\cos(x_{2})+\cos(x_{1})\sin(x_{2})}$

See also

• Identical function

Individual evidence

1. ^ Michael Merz: Mathematics for economists. Vahlen Verlag, 2012, ISBN 978-3-8006-4482-7 , Chapter 4.2: Equations. P. 71.
2. ↑ Wilfried Plassmann, Detlef Schulz (ed.): Formulas and tables of electrical engineering. Vieweg + Teubner-Verlag, 2014, ISBN 978-3-8348-0525-6 , Chapter 2.1: Types of equations.
3. ^ W. Busse v. Colbe, G. Laßmann: Business economics. Volume 1: Basics, production and cost theory. Springer Verlag, 1983, ISBN 3-540-16122-8 , Chapter 1.4.d: Identities (Identical Equations).
4. ↑ H. Geiger, K. Scheel (Ed.): Handbuch der Physik. Volume III: Mathematical Aids in Physics. Springer Verlag, 1928, Chapter 1.Ib: The concept of function.
5. ^ Adalbert Duschk: Lectures on higher mathematics. Springer-Verlag, Vienna 1949, ISBN 978-3-7091-3966-0 , § 8.3: Equation and identity.