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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. The equality between *A* and *B* is written *A* = *B*, and pronounced *A* equals *B*. The symbol "=" is called an "equals sign". Two objects that are not equal are said to be **distinct**.

- Etymology
- Basic properties
- Equality as predicate
- Identities
- Equations
- Congruences
- Approximate equality
- Relation with equivalence and isomorphism
- Logical definitions
- Equality in set theory
- Set equality based on first-order logic with equality
- Set equality based on first-order logic without equality
- See also
- Notes
- References
- External links

For example:

- means that x and y denote the same object.
^{ [1] } - means that, if x is any number, the two expressions have the same value. This may also be interpreted by saying that the two sides of the equals sign represent the same function.
- if and only if This assertion, which uses set-builder notation, means that, if the elements satisfying the property are the same as the elements satisfying then the two uses of the set-builder notation define the same set. This property often expressed as "two sets that have the same elements are equal." It is one of the usual axioms of set theory, called Axiom of extensionality.
^{ [2] }

The etymology of the word is from the Latin * aequālis * (“equal”, “like”, “comparable”, “similar”) from * aequus * (“equal”, “level”, “fair”, “just”).

*Substitution property*: For any quantities*a*and*b*and any expression*F*(*x*), if*a*=*b*, then*F*(*a*) =*F*(*b*) (if both sides make sense, i.e. are well-formed).

Some specific examples of this are:

- For any real numbers
*a*,*b*, and*c*, if*a*=*b*, then*a*+*c*=*b*+*c*(here*F*(*x*) is*x*+*c*); - For any real numbers
*a*,*b*, and*c*, if*a*=*b*, then*a*−*c*=*b*−*c*(here*F*(*x*) is*x*−*c*); - For any real numbers
*a*,*b*, and*c*, if*a*=*b*, then*ac*=*bc*(here*F*(*x*) is*xc*); - For any real numbers
*a*,*b*, and*c*, if*a*=*b*and*c*is not zero, then*a*/*c*=*b*/*c*(here*F*(*x*) is*x*/*c*).

- For any real numbers

*Reflexive property*: For any quantity*a*,*a*=*a*.

*Symmetric property*: For any quantities*a*and*b*, if*a*=*b*, then*b*=*a*.

These three properties make equality an equivalence relation. They were originally included among the Peano axioms for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be deduced from substitution and reflexive properties.

When *A* and *B* are not fully specified or depend on some variables, equality is a proposition, which may be true for some values and false for some other values. Equality is a binary relation or, in other words, a two-argument predicate, which may produce a truth value (*false* or *true*) from its arguments. In computer programming, its computation from two expressions is known as comparison.

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 (*x* + 1)^{2} = *x*^{2} + 2*x* + 1. Sometimes, but not always, an identity is written with a triple bar: (*x* + 1)^{2} ≡ *x*^{2} + 2*x* + 1.

An equation is a problem of finding values of some variables, called *unknowns*, for which the specified equality is true. *Equation* may also refer to an equality relation that is satisfied only for the values of the variables that one is interested in. For example, *x*^{2} + *y*^{2} = 1 is the *equation* of the unit circle.

There is no standard notation that distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantics of expressions and the context. An identity is *asserted* to be true for all values of variables in a given domain. An "equation" may sometimes mean an identity, but more often it *specifies* a subset of the variable space to be the subset where the equation is true.

In some cases, one may consider as **equal** two mathematical objects that are only equivalent for the properties that are considered. Particularly in the case of geometry, this is where two geometric shapes are said to be equal when one may be moved to coincide with the other. The word **congruence** is also used for this kind of 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.

The binary relation "is approximately equal" between real numbers or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small differences can add up to something big). However, equality almost everywhere *is* transitive.

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 *x ^{R}* the equivalence class of

In some contexts, equality is sharply distinguished from * equivalence * or * isomorphism.*^{ [3] } For example, one may distinguish * fractions * from * rational numbers,* the latter being equivalence classes of fractions: the fractions and 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

- and

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

However, there are other choices of isomorphism, such as

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.

Leibniz characterized the notion of equality as follows:

- Given any
*x*and*y*,*x*=*y*if and only if, given any predicate*P*,*P*(*x*) if and only if*P*(*y*).

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.

In first-order logic with equality, the axiom of extensionality states that two sets which *contain* the same elements are the same set.^{ [4] }

- Logic axiom:
*x*=*y*⇒ ∀*z*, (*z*∈*x*⇔*z*∈*y*) - Logic axiom:
*x*=*y*⇒ ∀*z*, (*x*∈*z*⇔*y*∈*z*) - Set theory axiom: (∀
*z*, (*z*∈*x*⇔*z*∈*y*)) ⇒*x*=*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."^{ [5] }

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.^{ [6] }

- Set theory definition: "
*x*=*y*" means ∀*z*, (*z*∈*x*⇔*z*∈*y*) - Set theory axiom:
*x*=*y*⇒ ∀*z*, (*x*∈*z*⇔*y*∈*z*)

- ↑ Rosser 2008 , p. 163.
- ↑ Lévy 2002 , pp. 13, 358. Mac Lane & Birkhoff 1999 , p. 2. Mendelson 1964 , p. 5.
- ↑ ( Mazur 2007 )
- ↑ Kleene 2002 , p. 189. Lévy 2002 , p. 13. Shoenfield 2001 , p. 239.
- ↑ Lévy 2002 , p. 4.
- ↑ Mendelson 1964 , pp. 159–161. Rosser 2008 , pp. 211–213

In mathematics, a **binary relation** over two sets *X* and *Y* is a set of ordered pairs (*x*, *y*) consisting of elements *x* in *X* and *y* in *Y*. That is, it is a subset of the Cartesian product *X* × *Y*. It encodes the information of relation: an element *x* is related to an element *y*, if and only if the pair (*x*, *y*) belongs to the set. Binary relation is the most studied form of relations among all n-ary relations.

**Naïve set theory** is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.

In mathematics, an **equivalence relation** is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:

In mathematics, when the elements of some set *S* have a notion of equivalence defined on them, then one may naturally split the set *S* into **equivalence classes**. These equivalence classes are constructed so that elements *a* and *b* belong to the same **equivalence class** if and only if they are equivalent.

**First-order logic**—also known as **predicate logic**, **quantificational logic**, and **first-order predicate calculus**—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as *Socrates is a man* one can have expressions in the form "there exists x such that x is Socrates and x is a man" and *there exists* is a quantifier while *x* is a variable. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

In mathematics, an **isomorphism** is a mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are **isomorphic** if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος *isos* "equal", and μορφή *morphe* "form" or "shape".

In mathematical logic, the **Peano axioms**, also known as the **Dedekind–Peano axioms** or the **Peano postulates**, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.

In mathematics, an **inequality** is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:

In mathematics, a binary relation *R* over a set *X* is **reflexive** if it relates every element of *X* to itself. Formally, this may be written ∀*x* ∈ *X* : *x R x*, or as I ⊆ *R* where I is the identity relation on *X*.

In mathematics, a homogeneous relation *R* over a set *X* is **transitive** if for all elements *a*, *b*, *c* in *X*, whenever *R* relates *a* to *b* and *b* to *c*, then *R* also relates *a* to *c*. Transitivity is a key property of both partial orders and equivalence relations.

In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. This sort of quantification is known as **uniqueness quantification** or **unique existential quantification**, and is often denoted with the symbols "∃!" or "∃_{=1}". For example, the formal statement

A set is **closed** under an operation if performance of that operation on members of the set always produces a member of that set. For example, the positive integers are closed under addition, but not under subtraction: is not a positive integer even though both 1 and 2 are positive integers. Another example is the set containing only zero, which is closed under addition, subtraction and multiplication.

In mathematics, there are several ways of defining the real number system as an ordered field. The *synthetic* approach gives a list of axioms for the real numbers as a *complete ordered field*. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense that there is a model for the axioms, and any two such models are isomorphic. Any one of these models must be explicitly constructed, and most of these models are built using the basic properties of the rational number system as an ordered field.

In mathematics, two sets or classes *A* and *B* are **equinumerous** if there exists a one-to-one correspondence between them, i.e. if there exists a function from *A* to *B* such that for every element *y* of *B* there is exactly one element *x* of *A* with *f*(*x*) = *y*. Equinumerous sets are said to have the same cardinality. The study of cardinality is often called **equinumerosity** (*equalness-of-number*). The terms **equipollence** (*equalness-of-strength*) and **equipotence** (*equalness-of-power*) are sometimes used instead.

**Tarski's axioms**, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory. Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms.

This article examines the implementation of mathematical concepts in set theory. The implementation of a number of basic mathematical concepts is carried out in parallel in ZFC and in NFU, the version of Quine's New Foundations shown to be consistent by R. B. Jensen in 1969.

In mathematics and abstract algebra, a **relation algebra** is a residuated Boolean algebra expanded with an involution called **converse**, a unary operation. The motivating example of a relation algebra is the algebra 2^{X²} of all binary relations on a set *X*, that is, subsets of the cartesian square *X*^{2}, with *R*•*S* interpreted as the usual composition of binary relations *R* and *S*, and with the converse of *R* as the converse relation.

In constructive mathematics, an **apartness relation** is a constructive form of inequality, and is often taken to be more basic than equality. It is often written as # to distinguish from the negation of equality ≠, which is weaker.

In mathematics, **Euclidean relations** are a class of binary relations that formalizes "Axiom 1" in Euclid's Elements: "Magnitudes which are equal to the same are equal to each other."

- Kleene, Stephen Cole (2002) [1967].
*Mathematical Logic*. Mineola, New York: Dover Publications. ISBN 978-0-486-42533-7.CS1 maint: ref=harv (link) - Lévy, Azriel (2002) [1979].
*Basic set theory*. Mineola, New York: Dover Publications. ISBN 978-0-486-42079-0.CS1 maint: ref=harv (link) - Mac Lane, Saunders; Birkhoff, Garrett (1999) [1967].
*Algebra*(Third ed.). Providence, Rhode Island: American Mathematical Society.CS1 maint: ref=harv (link) - Mazur, Barry (12 June 2007),
*When is one thing equal to some other thing?*(PDF) - Mendelson, Elliott (1964).
*Introduction to Mathematical Logic*. New York: Van Nostrand Reinhold.CS1 maint: ref=harv (link) - Rosser, John Barkley (2008) [1953].
*Logic for mathematicians*. Mineola, New York: Dover Publication. ISBN 978-0-486-46898-3.CS1 maint: ref=harv (link) - Shoenfield, Joseph Robert (2001) [1967].
*Mathematical Logic*(2nd ed.). A K Peters. ISBN 978-1-56881-135-2.CS1 maint: ref=harv (link)

- Hazewinkel, Michiel, ed. (2001) [1994], "Equality axioms",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4

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