Equality (math)

Equality , written as the equal sign " " in formulas , means complete agreement in mathematics . A math object is only equal to itself. There can of course be different names and descriptions for the same object, such as different arithmetic expressions for the same number, different definitions of the same geometrical figure, or different tasks that have the same unique solution. If one uses the mathematical formula language, such "designations and descriptions" are called terms . Which object is meant by a term depends on the context in which the term is "interpreted"; accordingly, a statement about the equality or inequality of two terms also depends on the context. What this relationship consists of is detailed in the section Equality within a set or structure . ${\ displaystyle =}$

What is the same is interchangeable. For example, if one knows that in a certain context for two terms and , then one can: ${\ displaystyle s = t}$ ${\ displaystyle s}$ ${\ displaystyle t}$

in a statement in which it appears as a component, one or more occurrences of replace by without changing the truth or falsity of the statement in the same context and ${\ displaystyle s}$ ${\ displaystyle s}$ ${\ displaystyle t}$
replace one or more occurrences of with in a term in which occurs as a component , whereby in the same context the modified term is the same as the original. ${\ displaystyle s}$ ${\ displaystyle s}$ ${\ displaystyle t}$

This principle “like can be replaced by like” is used, among other things, for algebraic transformations. If, for example, a term that is contained in another term or in a formula is simplified or calculated and the result is reinstated at the point of origin, this is an application of this principle, and likewise if the same operation is applied to both sides of an equation . Such transformations have been used to solve algebraic problems since ancient times, e.g. B. Diophant and al-Khwarizmi .

Objects that are indistinguishable and interchangeable in this way in every context are generally referred to as identical (or the same ), which says more than just the same (or the same ). There, but not in mathematics, equality means only a match in all relevant characteristics in the respective context, but not identity - a state of affairs that is called equivalence or congruence in mathematics , but not equality.

Equality is a fundamental concept in all of mathematics and is therefore not examined in the individual sub-areas of mathematics, but in mathematical logic . In contrast, the concept of identity is rarely used in mathematics in the sense of equality.

Equality within a set or structure

Mathematics deals with the relationship between mathematical objects within a set that is provided with a mathematical structure , but not with the nature of a mathematical object independent of the sets and structures to which it belongs. For this reason it is only a meaningful question in mathematics whether two objects from different sets are the same or different from one another if one set is part of the other or a higher-ranking set is needed. For example, whether the cardinal number 3 (in the sense of set theory: the thickness of a three-element set) is the same object as the real number 3 is only interesting if you want to build a structure in which cardinal numbers occur in the same context as real numbers - a unusual case in which one has to define how equality is meant.

But if one or more sets have been clearly defined, then it is clear what equality means: the elements of a set are only equal to themselves, and two sets are equal if and only if they contain the same elements. Building on this, one can form pairs and n -tuples with the help of the Cartesian product of sets, as well as functions that map a set into itself or into another set. The equality is transferred to such composite objects, whereby what is built up in the same way from the same components is the same.

The construction of the set of rational numbers from the set of whole numbers will serve as an example . At first glance, rational numbers are fractions of whole numbers with denominators other than zero, as a set of pairs of numbers . Then, however, the pairs and two different pairs would be two different rational numbers and , according to the definition of equality , and a definition according to which they should be equal would lead to a contradiction. How one can nevertheless arrive at a set of pairwise different rational numbers by forming equivalence classes is described in detail in the section Definition of the article Rational number . ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Z}}$ ${\ displaystyle \ mathbb {Z} \ times (\ mathbb {Z} \ setminus \ {0 \})}$ ${\ displaystyle \ langle 14.6 \ rangle}$ ${\ displaystyle \ langle 21.9 \ rangle}$ ${\ displaystyle {\ tfrac {14} {6}}}$ ${\ displaystyle {\ tfrac {21} {9}}}$

Statements about the rational numbers defined in this way can only be made if functions such as the arithmetic operations or relationships such as the smaller and larger relations are defined there. If this is not the case, there are at most statements about the equality of two differently written rational numbers, whose correctness or falsity is already determined by the definition of the rational numbers. In other words: the equality is indeed a relation on the set of rational numbers, but not one that could have been defined from there after the definition . Rather, it was created through the definition of with. ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle \ mathbb {Q}}$

Let us take the equation as an example of a statement about rational numbers . It only makes sense when it is known ${\ displaystyle (x + y) (xy) = x ^ {2} -y ^ {2}}$

what set of objects (here the set of rational numbers) we are talking about , ${\ displaystyle \ mathbb {Q}}$
how the arithmetic operations (here the four basic arithmetic operations - exponentiation with constant natural numbers is just an abbreviation for repeated multiplications) are defined on this basic set of objects and
for which elements of the set the occurring free variables (here that and that ) stand. ${\ displaystyle x}$ ${\ displaystyle y}$

These three things, namely the underlying set of objects, the definition of the functions and relations occurring on this set - but not that of the relation equality - as well as the assignment of the free variables with elements of the set thus form the context in which the statement is interpreted , that is clearly true or false. You can do this in a formal way as shown under interpretation , but even without such a formalism, every statement only has a meaning if these three components of the interpretation of the statement are fixed.

The assignment of the free variables usually results from the context. In this example - as with all formulas that can be found in formulas - it is usually meant that the statement is generally valid, i.e. H. it applies to everyone and from the basic set. In another context, the formula could also have represented the task of finding all for a given so that the equation is fulfilled (see equation ). Equations and other statements that contain free variables, about which nothing is specified in the context of use, can be generally valid, satisfiable or unsatisfiable, depending on whether they are true for all, some or no assignments of the free variables with elements of the basic set. ${\ displaystyle x}$ ${\ displaystyle y}$ ${\ displaystyle y}$ ${\ displaystyle x}$

The two other components of the interpretation, i.e. the basic set and the functions and relations defined on it, together form the mathematical structure in the context of which the statement is generally valid, satisfiable or unsatisfiable. In the structure, which consists of the basic arithmetic operations, the above equation is generally valid, in structures with non-commutative multiplication it is not, e.g. B. in the 2x2 matrices of integers with the usual matrix multiplication . ${\ displaystyle \ mathbb {Q}}$

Individual evidence

↑ Helmuth Gericke: Mathematics in antiquity and the Orient . 4th edition. Fourier, Wiesbaden 1996, ISBN 3-925037-64-0 , p. 144, 198 .

[1] Helmuth Gericke: Mathematics in antiquity and the Orient . 4th edition. Fourier, Wiesbaden 1996, ISBN 3-925037-64-0 , p. 144, 198 .