Equality (mathematics)

Equality, written in formulas as the equal sign " ", means complete agreement in mathematics. A mathematical object is equal only to itself. Of course, there can be different names and descriptions for the same object, such as different arithmetic expressions for the same number, different definitions of the same geometric figure, or different problems 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 context consists of is explained in detail in the section Equality within a set or structure.

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

in a statement in which occurs as a constituent, replace one or more occurrences of with without changing the truth or falsity of the statement in the same context; and
in a term in which occurs as a constituent, replace one or more occurrences of with , where in the same context the modified term is the same as the original term.

This principle "like may be replaced by like" is used, among other things, in algebraic transformations. If, for example, a term contained in another term or in a formula is simplified or calculated and the result is inserted again at the place 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 since ancient times to solve algebraic problems, e.g. by Diophant and by al-Chwarizmi.

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

Equality is a fundamental concept in all of mathematics and is therefore not studied in the individual subfields of mathematics, but in mathematical logic. The concept of identity, on the other hand, 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 provided with a mathematical structure, but not with the nature of a mathematical object independent of the sets and structures to which it belongs. Therefore, it is a meaningful question in mathematics whether two objects from different sets are the same or different from each other only if one set is part of the other or a set superior to both is needed. For example, whether the cardinal number 3 (in the sense of set theory: the power of a three-element set) is the same object as the real number 3 is only interesting if one wants to build a structure in which cardinal numbers occur next to real numbers in the same context - an unusual case in which one has to define how the equality is meant.

But if one or more sets are defined unambiguously, it is clear what equality means: the elements of a set are equal only to themselves, and two sets are equal if they contain the same elements. Based on this, one can form pairs and n-tuples using the Cartesian product of sets, as well as functions that map a set into itself or into another set. Equality transfers to such composite objects, where equal is what is built up in the same way from the same components.

As an example, consider the construction of the set $\mathbb {Q}$ of the rational numbers from the set $\mathbb {Z}$ of the integers serve. Rational numbers are, at first sight, fractions of integers with nonzero denominators, conceived as a set of pairs of numbers from $\mathbb {Z} \times (\mathbb {Z} \setminus \{0\})$ . But then the pairs ⟨ $\langle 14,6\rangle$ and ⟨ would be $\langle 21,9\rangle$ two different pairs, hence, according to the definition of equality, two different rational numbers ${\tfrac {14}{6}}$ and ${\tfrac {21}{9}}$ , and a definitional specification that they should be equal would lead to a contradiction. How one nevertheless arrives at a set of pairwise different rational numbers by forming equivalence classes is described in detail in the Definition section of the article Rational Number.

Statements about the rational numbers defined in such a way can be made only if functions are defined there as for example the arithmetic operations or relations as for example the smaller and larger relation. If one does not have this, there are at most statements about the equality of two differently written rational numbers, whose correctness or incorrectness is already determined on the basis of the definition of the rational numbers. In other words, equality is indeed a relation on the set $\mathbb {Q}$ of the rational numbers, but not one that could have been defined there after the definition of could still have been defined there. Rather, it arose $\mathbb {Q}$ from the definition of came into being with. $\mathbb {Q}$

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

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

These three things, namely the underlying set of objects, the definition of the occurring functions and relations on this set - but not that of the relation equality - as well as the occupation of the free variables with elements of the set thus form the context in which the statement is interpreted, i.e. is true or false in an unambiguous way. This can be done in a formal way as shown under Interpretation, but even without such formalism every statement has a meaning only 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 found in formula collections - it is usually meant that the statement is generally valid, i.e. it applies to all and from the basic set. In another context, the formula could have represented the task of finding all for given such that the equation is satisfied (see Equation). Equations and other statements containing free variables about which nothing is specified in the context of use may be generally valid, satisfiable, or unsatisfiable, depending on whether they are true for all, some, or no assignments of the free variables to elements of the basic set.

The other two 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 consisting of $\mathbb {Q}$ with the basic arithmetic operations, the said equation is universally valid; in structures with noncommutative multiplication it is not, for example, in the 2x2 matrices of integers with the usual matrix multiplication.

Questions and Answers

Q: What is the symbol used to represent equality in mathematics?

A: The equals sign (=) is used to represent equality in mathematics.

Q: How can two mathematical objects be equivalent?

A: Two mathematical objects can be equivalent if they are related by an equivalence relation. This is often represented using symbols such as ∼ or ≡.

Q: What does it mean when two expressions denote equal quantities?

A: When two expressions denote equal quantities, it means that they are equal and this statement is referred to as an equation or an equality.

Q: How do mathematicians differentiate between equations and inequalities?

A: Equations are equal while inequalities are unequal.

Q: What is the difference between congruence and similarity in geometry?

A: Congruence occurs when one geometrical object can be moved or rotated so that it fits exactly where the other one is, without shrinking or enlarging either of them. Similarity occurs when shrinking or enlarging one of the two objects is needed for them to fit together. The congruence relation is often represented by the symbol ≅ while the similarity relation is represented by the symbol ∼ .

Q: In computer science, what operator compares actual values of objects rather than where variables point to?

A: In computer science, languages which have pointers usually use another operator (such as Java's 'equals' method) which compares actual values of objects rather than where variables point to.

Q: How do people define equality in social sciences?

A: In social sciences, two people are considered equal if many of the same things are true about them, such as having similar levels of education and money and being around the same age. Another name for a person who is equal to another person in this sense would be a peer.