Equality (mathematics) explained

In mathematics, equality is a relationship between two quantities or, more generally, two mathematical expressions, asserting that the quantities have the same value, or that the expressions represent the same mathematical object. Equality between and is written, and pronounced " equals ". In this equality, and are the members of the equality and are distinguished by calling them left-hand side or left member, and right-hand side or right member. Two objects that are not equal are said to be distinct.

A formula such as

x=y,

where and are any expressions, means that and denote or represent the same object.[1] For example,

1.5=3/2,

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

\{x\midx\in\Zand0<x\le3\}=\{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".[2])

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=x2+2x+1,

means that if is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function (equality of functions), or that the two expressions denote the same polynomial (equality of polynomials).[3] [4]

Etymology

The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just").[5]

Basic properties

for every, one has .

for every and, if, then .

for every,, and, if and, then .[6] [7]

Informally, this just means that if, then can replace in any mathematical expression or formula.

f(x)

, if, then

f(a)=f(b)

.[8]
For example:

2a-5=2b-5

. (Here,

f(x)=2x-5

. A unary operation)

a2=2b2

, then

a2/b2=2

. (Here,

f(x,b)=x/b2

. A binary operation)

g(a)=h(a)

, then \fracg(a) = \frach(a). (Here, f(x) = \frac. An operation over functions (i.e. an operator), called the derivative)

S

, those first three properties make equality an equivalence relation on

S

. In fact, equality is the unique equivalence relation on

S

whose equivalence classes are all singletons.

Equality as predicate

In logic, a predicate is a proposition which may have some free variables. When A and B are not fully specified or depend on some variables, equality is a predicate, which may be true for some values and false for other values. Equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value (true or false) from its arguments. In computer programming, equality is called a Boolean-valued expression, and its computation from the two expressions is known as comparison.

Identities

See main article: Identity (mathematics). When A and B may be viewed as functions of some variables, then A = B means that A and B define the same function. Such an equality of functions is sometimes called an identity. An example is

\left(x+1\right)\left(x+1\right)=x2+2x+1.

Sometimes, but not always, an identity is written with a triple bar:

\left(x+1\right)\left(x+1\right)\equivx2+2x+1.

[9]

Equations

An equation is the problem of finding values of some variable, called, for which the specified equality is true. Each value of the unknown for which the equation holds is called a of the given equation; also stated as the equation. For example, the equation

x2-6x+5=0

has the values

x=1

and

x=5

as its only solutions. The terminology is used similarly for equations with several unknowns.[10]

An equation can be used to define a set. For example, the set of all solution pairs

(x,y)

of the equation

x2+y2=1

forms the unit circle in analytic geometry; therefore, this equation is called .

An identity is an equality that is true for all values of its variables in a given domain.[11] An "equation" may sometimes mean an identity, but more often than not, it a subset of the variable space to be the subset where the equation is true. 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.[12]

See also: Equation solving

In logic

In mathematical logic and mathematical philosophy, equality is often described through the following properties:[13] [14] [15]

\foralla(a=a)

a

,

a=a

. It is the first of the historical three laws of thought.

(a=b)\impliesl[\phi(a)\phi(b)r]

\phi(x),

(with a free variable), if

a=b

, then

\phi(a)

implies

\phi(b)

.

For example: For all real numbers and, if, then implies (here,

\phi(x)

is)

These properties offer a formal reinterpretation of equality from how it is defined in standard Zermelo–Fraenkel set theory (ZFC) or other formal foundations. In ZFC, equality only means that two sets have the same elements. However, mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that the expression "

1\cup2

" (see union) is an abuse of notation or meaningless. This is a more abstracted framework which is grounded in ZFC (that is, both axioms can be proved within ZFC), but is closer to how most mathematicians use equality.

Note that this says "Equality implies these two properties" not that "These properties define equality"; this is intentional. This makes it an incomplete axiomatization of equality. That is, it does not say what equality is, only what "equality" 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.[16] (See the following subsection)

If these properties 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. The converse of the Substitution property is the identity of indiscernibles, which states that two distinct things cannot have all their properties in common. 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 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.[17] This is why the properties are said to not form a complete axiomatization.

However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC.

These are sometimes taken as the definition of equality, such as in some areas of first-order logic.[18]

Derivations of basic properties

xRy\Leftrightarrowx=y

), assume

a\inS

. Then

a=a

by the Law of identity, thus

aRa

.

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, many mathematicians refer to both as "Reflexivity", which is generally harmless.[19]

xRy\Leftrightarrowx=y

), assume there are elements

a,b\inS

such that

aRb

. Then, take the formula

\phi(x):xRa

. So we have

(a=b)\implies(aRabRa)

. Since

a=b

by assumption, and

aRa

by Reflexivity, we have that

bRa

.

xRy\Leftrightarrowx=y

), assume there are elements

a,b,c\inS

such that

aRb

and

bRc

. Then take the formula

\phi(x):xRc

. So we have

(b=a)\implies(bRcaRc)

. Since

b=a

by symmetry, and

bRc

by assumption, we have that

aRc

.

f(x)

, assume there are elements and from its domain such that, then take the formula

\phi(x):f(a)=f(x)

. So we have

(a=b)\implies[(f(a)=f(a))(f(a)=f(b))]


Since

a=b

by assumption, and

f(a)=f(a)

by reflexivity, we have that

f(a)=f(b)

.

This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above.

Approximate equality

There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers, defined by formulas 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).

The binary relation "is approximately equal" (denoted by the symbol

) between real numbers or other things, even if more precisely defined, is not transitive (since many small differences can add up to something big). However, equality almost everywhere is transitive.

A questionable equality under test may be denoted using the

\stackrel{?}{=}

symbol.[20]

Relation with equivalence, congruence, and isomorphism

See main article: Equivalence relation, Isomorphism, Congruence relation and Congruence (geometry).

Viewed as a relation, equality is the archetype of the more general concept of an equivalence relation on a set: those binary relations that are reflexive, symmetric and transitive. The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by xR the equivalence class of x, consisting of all elements z such that x R z. Then the relation x R y is equivalent with the equality xR = yR. It follows that equality is the finest equivalence relation on any set S in the sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element).

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

\{A,B,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

A\mapsto1,B\mapsto2,C\mapsto3.

However, there are other choices of isomorphism, such as

A\mapsto3,B\mapsto2,C\mapsto1,

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.[21] [22] [23]

Equality in set theory

See main article: Axiom of extensionality.

Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality.

Set equality based on first-order logic with equality

In first-order logic with equality, the axiom of extensionality states that two sets which contain the same elements are the same set.[24]

x=y\implies\forallz,(z\inx\iffz\iny)

x=y\implies\forallz,(x\inz\iffy\inz)

(\forallz,(z\inx\iffz\iny))\impliesx=y

Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by 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."[25]

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.[26]

(x=y):=\forallz,(z\inx\iffz\iny)

x=y\implies\forallz,(x\inz\iffy\inz)

See also

References

Notes and References

  1. .
  2. . . .
  3. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
  4. Pratt, Vaughan, "Algebra", The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL: https://plato.stanford.edu/entries/algebra/#Laws
  5. Web site: Definition of EQUAL . 2020-08-09 . . en . 2020-09-15 . https://web.archive.org/web/20200915001915/https://www.merriam-webster.com/dictionary/equal . live .
  6. Book: Stoll, Robert R.. 1963. Set Theory and Logic. Dover Publications. San Francisco, CA. 978-0-486-63829-4.
  7. Book: 3-87144-118-X . Lilly Görke . Mengen  - Relationen  - Funktionen . Zürich . Harri Deutsch . 4th . 1974 . Here: sect.3.5, p.103.
  8. Equality axioms. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
  9. Web site: Identity – math word definition – Math Open Reference. www.mathopenref.com. 2019-12-01.
  10. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
  11. Equation. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equation&oldid=32613
  12. Web site: Marcus . Solomon . Watt . Stephen M. . What is an Equation? . 2019-02-27 . Solomon Marcus .
  13. Equality axioms. Springer Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
  14. Deutsch, Harry and Pawel Garbacz, "Relative Identity", The Stanford Encyclopedia of Philosophy (Fall 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), forthcoming URL: https://plato.stanford.edu/entries/identity-relative/#StanAccoIden
  15. Forrest, Peter, "The Identity of Indiscernibles", The Stanford Encyclopedia of Philosophy (Winter 2020 Edition), Edward N. Zalta (ed.), URL: https://plato.stanford.edu/entries/identity-indiscernible/#Form
  16. Equality axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
  17. Encyclopedia: French . Steven . 2019 . Identity and Individuality in Quantum Theory . Stanford Encyclopedia of Philosophy . 1095-5054 .
  18. [Melvin Fitting|Fitting, M.]
  19. Equality axioms. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equality_axioms&oldid=46837
  20. Web site: Find all Unicode Characters from Hieroglyphs to Dingbats – Unicode Compart .
  21. Eilenberg . S. . Mac Lane . S. . 1942 . Group Extensions and Homology . Annals of Mathematics . 43 . 4 . 757–831 . registration . 10.2307/1968966 . 1968966 . 0003-486X . JSTOR.
  22. Web site: Marquis . Jean-Pierre . 2019 . Category Theory . . . 26 September 2022.
  23. Book: Hofmann . Martin . Twenty Five Years of Constructive Type Theory . Streicher . Thomas . 1998 . Clarendon Press . 978-0-19-158903-4 . Sambin . Giovanni . Oxford Logic Guides . 36 . 83–111 . The groupoid interpretation of type theory . 1686862 . Thomas Streicher . Smith . Jan M. . https://books.google.com/books?id=pLnKggT_In4C&pg=PA83.
  24. . . .
  25. .
  26. .