Equality (mathematics)

This article is about the concept of equality in mathematics. For the symbol, see equals sign.

Equals sign

In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object. Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS). Two objects that are not equal are said to be distinct.

A formula such as $x=y,$ where x and y are any expressions, means that x and y denote or represent the same object. For example,

$1.5=3/2,$

are two notations for the same number. Similarly, using set builder notation,

$\{x\mid x\in \mathbb {Z} {\text{ and }}0<x\leq 3\}=\{1,2,3\},$

since the two sets have the same elements. (This equality results from the axiom of extensionality that is often expressed as "two sets that have the same elements are equal".)

The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if the members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols.

An identity, such as $(x+1)^{2}=x^{2}+2x+1,$ means that if x is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two expressions represent the same function (equality of functions).

Etymology

The first use of an equals sign, equivalent to $14x+15=71$ in modern notation. From The Whetstone of Witte (1557) by Robert Recorde.

Recorde's introduction of ⟨=⟩

The word equal is derived from the Latin aequālis ('like', 'comparable', 'similar'), which itself stems from aequus ('level', 'just'). The word entered Middle English around the 14th century, borrowed from Old French equalité (modern égalité).

The equals sign ⟨=⟩, now universally accepted in mathematics for equality, was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte (1557). The original form of the symbol was much wider than the present form. In his book, Recorde explains his design of the "Gemowe lines", from the Latin gemellus ('twin'), using two parallel lines to represent equality because he believed that "no two things could be more equal." Later, a vertical version ⟨||⟩ was also used by some but never overtook Recorde's version.

It was common into the 18th century to use an abbreviation of the word equals as the symbol for equality; examples included ⟨æ⟩ and ⟨œ⟩, from the Latin aequālis. Diophantus's use of ⟨ἴσ⟩, short for ἴσος (ísos 'equals'), in Arithmetica (c. 250 AD) is considered one of the first uses of an equals sign.

Basic properties

• Reflexivity: for every a, one has a = a.
• Symmetry: for every a and b, if a = b, then b = a.
• Transitivity: for every a, b, and c, if a = b and b = c, then a = c.
• Substitution: Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning. (For a formal explanation, see § In logic)
  For example:
  • Given real numbers a and b, if a = b, then $a>0$ implies $b>0$
• Operation application: for every a and b, with some operation $f(x)$, if a = b, then $f(a)=f(b)$.
  For example:
  • Given real numbers a and b, if a = b, then $2a-5=2b-5$. (Here, $f(x)=2x-5$. A unary operation)
  • Given positive reals a and b, if $a^{2}=2b^{2}$, then $a^{2}/b^{2}=2$. (Here, $f(x,y)=x/y^{2}$ at $y=b$. A binary operation)
  • Given real functions ⁠$g$⁠ and ⁠$h$⁠ over some variable a, if $g(a)=h(a)$ for all a, then ${\textstyle {\frac {d}{da}}g(a)={\frac {d}{da}}h(a)}$ for all a. (Here, ${\textstyle f(x)={\frac {dx}{da}}}$. An operation over functions (i.e. an operator), called the derivative).

The first three properties are generally attributed to Giuseppe Peano for being the first to explicitly state these as fundamental properties of equality in his Arithmetices principia (1889). However, the basic notions have always existed; for example, in Euclid's Elements (c. 300 BC), he includes 'common notions': "Things that are equal to the same thing are also equal to one another" (transitivity), "Things that coincide with one another are equal to one another" (reflexivity), along with some operation-application properties for addition and subtraction. The substitution property is genreally attributed to Gottfried Leibniz.

Equations

An equation is a symbolic equality of two mathematical expressions connected with an equals sign (=). Equation solving is the problem of finding values of some variable, called unknown, for which the specified equality is true. Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying the equation. For example, the equation $x^{2}-6x+5=0$ has the values $x=1$ and $x=5$ as its only solutions. The terminology is used similarly for equations with several unknowns.

In mathematical logic and computer science, an equation may described as a binary formula or Boolean-valued expression, which may be true for some values of the variables (if any) and false for other values. More specifically, an equation represents a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments. In computer programming, the computation from the two expressions is known as comparison.

An equation can be used to define a set. For example, the set of all solution pairs $(x,y)$ of the equation $x^{2}+y^{2}=1$ forms the unit circle in analytic geometry; therefore, this equation is called the equation of the unit circle.

Identities

An identity is an equality that is true for all values of its variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it specifies a subset of the variable space to be the subset where the equation is true. An example is $\left(x+1\right)\left(x+1\right)=x^{2}+2x+1$, which is true for each real number $x$. There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context. Sometimes, but not always, an identity is written with a triple bar: $\left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.$

Definitions

Equality and equations are often used to introduce new terms or symbols, establish equivalences, and introduce shorthand for complex expressions. When defining a new symbol, it is usually denoted with ($:=$). It is similar to the concept of assignment of a variable in computer science. For example, ${\textstyle \mathbb {e} :=\sum _{n=0}^{\infty }{\frac {1}{n!}}}$ defines Euler's number, and $i^{2}=-1$ is the defining property of the imaginary number $i$.

In mathematical logic, this is called an extension by definition (by equality) which is a conservative extension to a formal system. This is done by taking the equation defining the new constant symbol as a new axiom of the theory.

The first recorded symbolic use of "Equal by definition" appeared in Logica Matematica (1894) by Cesare Burali-Forti, an Italian mathematician. Burali-Forti, in his book, used the notation ($=_{\text{Def}}$).

In logic

Gottfried Leibniz, a major contributor to 17th-century mathematics and philosophy of mathematics, and whom the Substitution property of equality is named after.

Equality (or identity) is often considered a primitive notion, informally said to be "a relation each thing bears to itself and to no other thing". This characterization is notably circular (“no other thing”) and paradoxical too, unless the notion of "each thing" is qualified. Around the 17th century, with the growth of modern logic, it became necessary to have a more concrete notion of equality. In foundations of mathematics, especially mathematical logic and analytic philosophy, equality is often axiomatized through the following properties:

Axioms

$\forall a(a=a)$

• Law of identity: Stating that each thing is identical with itself, without restriction. That is, for every $a$, $a=a$. It is the first of the traditional three laws of thought.

$(a=b)\implies {\bigl [}\phi (a)\Rightarrow \phi (b){\bigr ]}$

• Substitution property: Sometimes referred to as Leibniz's law, generally states that if two things are equal, then any property of one must be a property of the other. It can be stated formally as: for every a and b, and any formula $\phi (x),$ (with a free variable x), if $a=b$, then $\phi (a)$ implies $\phi (b)$.
  For example:
  • For all real numbers a and b, if a = b, then a ≥ 0 implies b ≥ 0 (here, $\phi (x)$ is x ≥ 0).

Function application is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms. (See § Derivations of basic properties)

The Law of identity is distinct from reflexivity in two main ways: first, the Law of Identity applies only to cases of equality, and second, it is not restricted to elements of a set. However, in mathematics and logic, both are often referred to as "Reflexivity", which is generally harmless.

This says "Equality implies these two properties" not that "These properties define equality". This makes it an incomplete axiomatization of equality. That is, it does not say what equality is, only what an "equality relation" must satify. However, the two axioms as stated are still generally useful, even as an incomplete axiomatization of equality, as they are usually sufficient for deducing most properties of equality that mathematicians care about. (See § Derivations of basic properties)

In first-order logic, these are axiom schemas, each of which specifies an infinite set of axioms. If a theory has a binary formula which satisfies Law of Identity and Substitution, it is common to say that has an equality, or is a theory with equality. It is possible to define equality within the theorem in terms of the relations, by letting $\phi$ range through the possible formulas, this is called extensionality. In this way, the equality relation may now be interpreted by an arbitrary equivalence relation on the domain. These axioms are useful in first-order logic, especially in automated theorem proving.

Objections

As mentioned above, these axioms don't explicitly define equality, in the sense that we still don't know if two objects are equal, only that if they're equal, then they have the same properties. If these axioms were to define a complete axiomatization of equality, meaning, if they were to define equality, then the converse of the second statement must be true. This is because any reflexive relation satisfying the substitution property within a given theory would be considered an "equality" for that theory. The converse of the Substitution property is the identity of indiscernibles, which states that two distinct things cannot have all their properties in common. Stated symbolically as:

$\forall \phi {\bigl [}\phi (a)\Leftrightarrow \phi (b){\bigr ]}\implies (a=b)$

In mathematics, the identity of indiscernibles is usually rejected since indiscernibles in mathematical logic are not necessarily forbidden. Set equality in ZFC is capable of declairing these indiscernibles as not equal, but an equality solely defined by these properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of pure math, the identity of indiscernibles has attracted much controversy and criticism, especially from corpuscular philosophy and quantum mechanics. This is why the properties are said to not form a complete axiomatization.

Extensionality

Main article: Extensionality

Extensionality is an axiom that defines objects of a certain kind to be equal by their external properties or relationships, rather than their intrinsic nature or internal construction. For example, in set theory an axiom of extensionality defined two sets to be equal if and only if they have the same elements. Similarly, two functions may be defined to be equal if they return the same outputs for all inputs, regardless of how they are defined or computed, often called an identity. Each of these may be stated symbolically as:

$\forall A\forall B(A=B\iff \forall x(x\in A\iff x\in B))$

$\forall f\forall g(f=g\iff \forall x(f(x)=g(x)\,))$

Derivations of basic properties

• Reflexivity of Equality: Given some set S with a relation R induced by equality ($xRy\Leftrightarrow x=y$), assume $a\in S$. Then $a=a$ by the Law of identity, thus $aRa$.

• Symmetry of Equality: Given some set S with a relation R induced by equality ($xRy\Leftrightarrow x=y$), assume there are elements $a,b\in S$ such that $aRb$. Then, take the formula $\phi (x):xRa$. So we have $(a=b)\implies (aRa\Rightarrow bRa)$. Since $a=b$ by assumption, and $aRa$ by Reflexivity, we have that $bRa$.
• Transitivity of Equality: Given some set S with a relation R induced by equality ($xRy\Leftrightarrow x=y$), assume there are elements $a,b,c\in S$ such that $aRb$ and $bRc$. Then take the formula $\phi (x):xRc$. So we have $(b=a)\implies (bRc\Rightarrow aRc)$. Since $b=a$ by symmetry, and $bRc$ by assumption, we have that $aRc$.
• Function application: Given some function $f(x)$, assume there are elements a and b from its domain such that a = b, then take the formula $\phi (x):f(a)=f(x)$. So we have
  $(a=b)\implies [(f(a)=f(a))\Rightarrow (f(a)=f(b))]$
  Since $a=b$ by assumption, and $f(a)=f(a)$ by reflexivity, we have that $f(a)=f(b)$.

In set theory

Main article: Axiom of extensionality

Two sets of polygons in Euler diagrams. These sets are equal since both have the same elements, even though the arrangement differs.

Set theory is the branch of mathematics that studies sets, which can be informally described as "collections of objects." Although objects of any kind can be collected into a set, set theory – as a branch of mathematics – is mostly concerned with those that are relevant to mathematics as a whole. Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set). In a formalized set theory, this is usually defined by an axiom called the Axiom of extensionality

For example, using set builder notation,

$\{x\mid x\in \mathbb {Z} {\text{ and }}0<x\leq 3\}=\{1,2,3\},$

Which states that "The set of all integers greater than 0 but not more than 3 is equal to the set containing only 1, 2, and 3", dispite the differences in notation.

Background

Ernst Zermelo, a contributer to modern Set theory, was the first to explicitly formalize set equality in his Zermelo set theory (now obsolete), by his Axiom der Bestimmtheit.

Around the turn of the 20th century, mathematics faced several paradoxes and counter-intuitive results. For example, Russell's paradox showed a contradiction of naive set theory, it was shown that the parallel postulate cannot be proved, the existence of mathematical objects that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with Peano arithmetic. The result was a foundational crisis of mathematics.

The resolution of this crisis involved the rise of a new mathematical discipline called mathematical logic, which studied formal logic within mathematics. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of axiomatic method and on set theory, specifically Zermelo–Fraenkel set theory, developed by Ernst Zermelo and Abraham Fraenkel. This set theory is now considered the most common foundation of mathematics.

Set equality based on first-order logic with equality

In first-order logic with equality (See § In logic), the axiom of extensionality states that two sets which contain the same elements are the same set.

• Logic axiom: $x=y\implies \forall z,(z\in x\iff z\in y)$
• Logic axiom: $x=y\implies \forall z,(x\in z\iff y\in z)$
• Set theory axiom: $(\forall z,(z\in x\iff z\in y))\implies x=y$

The first two are given by the substitution property of equality from first-order logic; the last is a new axiom of the theory. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Azriel Lévy.

"The reason why we take up first-order predicate calculus with equality is a matter of convenience; by this we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic."

Set equality based on first-order logic without equality

In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in the same sets.

• Set theory definition: $(x=y)\ :=\ \forall z,(z\in x\iff z\in y)$
• Set theory axiom: $x=y\implies \forall z,(x\in z\iff y\in z)$

Proof of basic properties

• Reflexivity: Given a set $X$, assume $z\in X$, it follows trivially that $z\in X$, and the same follows in reverse, therefore $\forall z,(z\in X\iff z\in X)$ , thus $X=X$.
• Symmetry: Given sets $X,Y$, such that $X=Y$, then $\forall z,(z\in X\iff z\in Y)$, which implies $\forall z,(z\in Y\iff z\in X)$, thus $Y=X$.
• Transitivity: Given sets $X,Y,Z$, such that (1) $X=Y$ and (2) $Y=Z$, assume $z\in X$, then $z\in Y$ by (1), which implies $z\in Z$ by (2), and similarly for the reverse, therefore $\forall z,(z\in X\iff z\in Z)$, thus $X=Z$.

Similar relations

Approximate equality

Main article: Approximation § Mathematics

The sequence given by the perimeters of regular n-sided polygons that circumscribe the unit circle approximates $2\pi$

Numerical approximation is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis.

Calculations are likely to involve rounding errors and other approximation errors. Log tables, slide rules and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results.

If viewed as a binary relation, (denoted by the symbol $\approx$) between real numbers or other things, if precisely defined, is not an equivalence relation since it's not transitive, even if modeled as a fuzzy relation.

In computer science, equality is given by some relational operator. Real numbers are often approximated by floating-point numbers (A sequence of some fixed number of digits of a given base, scaled by an integer exponent of that base), thus it is common to store an expression that denotes the real number as to not lose precision. However, the equality of two real numbers given by an expression is known to be undecidable (specifically, real numbers defined by expressions involving the integers, the basic arithmetic operations, the logarithm and the exponential function). In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem).

A questionable equality under test may be denoted using the ${\stackrel {?}{=}}$ symbol.

Equivalence relation

Main article: Equivalence relation

Graph of an example equivalence with 7 classes

An equivalence relation is a mathematical relation that generalizes the idea of similarity or sameness. It is defined on a set $X$ as a binary relation $\sim$ that satisfies the three properties: reflexivity, symmetry, and transitivity. Reflexivity means that every element in $X$ is equivalent to itself ($a\sim a$ for all $a\in X$). Symmetry requires that if one element is equivalent to another, the reverse also holds ($a\sim b\implies b\sim a$). Transitivity ensures that if one element is equivalent to a second, and the second to a third, then the first is equivalent to the third ($a\sim b$ and $b\sim c\implies a\sim c$). These properties are enough to partition a set into disjoint equivilence classes. Conversley, every partition defines an equivalence class.

The equivalence relation of equality is a special case, as, if restricted to a given set $S$, it is the strictest possible equivalence relation on $S$; specifically, equality partitions a set into equivalence classes consisting of all singleton sets. Other equivalence relations, while less restrictive, often generalize equality by identifying elements based on shared properties or transformations, such as congruence in modular arithmetic or similarity in geometry.

Congrunce relation

Main article: Congruence relation

In abstract algebra, a congruence relation extends the idea of an equivalence relation to include the operation-application property. That is, given a set $X$, and a set of operations on $X$, then a congruence relation $\sim$ has the property that $a\sim b\implies f(a)\sim f(b)$ for all operations $f$ (here, written as unary to avoid cumbersome notation, but $f$ may be of any arity). A congruence relation on an algebraic structure such as a group, ring, or module is an equivalence relation that respects the operations defined on that structure.

Isomorphism

Main article: Isomorphism

In some contexts, equality is sharply distinguished from equivalence or isomorphism. For example, one may distinguish fractions from rational numbers, the latter being equivalence classes of fractions: the fractions $1/2$ and $2/4$ are distinct as fractions (as different strings of symbols) but they "represent" the same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set.

Similarly, the sets

$\{{\text{A}},{\text{B}},{\text{C}}\}$ and $\{1,2,3\}$

are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between them. For example

${\text{A}}\mapsto 1,{\text{B}}\mapsto 2,{\text{C}}\mapsto 3.$

However, there are other choices of isomorphism, such as

${\text{A}}\mapsto 3,{\text{B}}\mapsto 2,{\text{C}}\mapsto 1,$

and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism, is of fundamental importance in category theory and is one motivation for the development of category theory.

In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and the associated symbol $\cong$) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects.

In geometry for instance, two geometric shapes are said to be equal or congruent when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly to isomorphisms of sets, the difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of category theory, as well as for homotopy type theory and univalent foundations.

See also

• Equipollence (geometry)
• Extensionality
• Glossary of mathematical symbols § Equality, equivalence and similarity
• Homotopy type theory
• Inequality
• Logical equality
• Logical equivalence
• Proportionality (mathematics)
• Relational operator § Equality

References

• Kleene, Stephen Cole (2002) [1967]. Mathematical Logic. Mineola, New York: Dover Publications. ISBN 978-0-486-42533-7.
• Lévy, Azriel (2002) [1979]. Basic set theory. Mineola, New York: Dover Publications. ISBN 978-0-486-42079-0.
• Mac Lane, Saunders; Birkhoff, Garrett (1999) [1967]. Algebra (Third ed.). Providence, Rhode Island: American Mathematical Society.
• Mazur, Barry (12 June 2007). When is one thing equal to some other thing? (PDF). Archived from the original (PDF) on 24 October 2019. Retrieved 13 December 2009.
• Mendelson, Elliott (1964). Introduction to Mathematical Logic. New York: Van Nostrand Reinhold.
• Rosser, John Barkley (2008) [1953]. Logic for mathematicians. Mineola, New York: Dover Publication. ISBN 978-0-486-46898-3.
• Shoenfield, Joseph Robert (2001) [1967]. Mathematical Logic (2nd ed.). A K Peters. ISBN 978-1-56881-135-2.