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35-514: [REDACTED] Look up equal , equals , equaling , or equaled in Wiktionary, the free dictionary. Equal ( s ) may refer to: Mathematics [ edit ] Equality (mathematics) . Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment [ edit ] Equals (film) , a 2015 American science fiction film Equals (game) ,

70-543: A and b , if a = b , then a ≥ 0 implies b ≥ 0 (here, ϕ ( x ) {\displaystyle \phi (x)} is x ≥ 0 ) 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, outside of set theory , mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that

105-578: 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

140-678: A 2022 song by J-Hope from Jack in the Box Other uses [ edit ] Equal (sweetener) , a brand of artificial sweetener. EQUAL Community Initiative , an initiative within the European Social Fund of the European Union. See also [ edit ] Equality (disambiguation) Equalizer (disambiguation) Equalization (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

175-485: A 2022 song by J-Hope from Jack in the Box Other uses [ edit ] Equal (sweetener) , a brand of artificial sweetener. EQUAL Community Initiative , an initiative within the European Social Fund of the European Union. See also [ edit ] Equality (disambiguation) Equalizer (disambiguation) Equalization (disambiguation) Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

210-452: A board game The Equals , a British pop group formed in 1965 "Equal", a 2016 song by Chrisette Michele from Milestone "Equal", a 2022 song by Odesza featuring Låpsley from The Last Goodbye "Equals", a 2009 song by Set Your Goals from This Will Be the Death of Us Equal (TV series) , a 2020 American docuseries on HBO = (album) , a 2021 album by Ed Sheeran "=",

245-400: A board game The Equals , a British pop group formed in 1965 "Equal", a 2016 song by Chrisette Michele from Milestone "Equal", a 2022 song by Odesza featuring Låpsley from The Last Goodbye "Equals", a 2009 song by Set Your Goals from This Will Be the Death of Us Equal (TV series) , a 2020 American docuseries on HBO = (album) , a 2021 album by Ed Sheeran "=",

280-425: 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

315-477: 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 . 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

350-446: 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 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 x  =  y . It follows that equality is the finest equivalence relation on any set S in

385-447: Is ( x + 1 ) ( x + 1 ) = x 2 + 2 x + 1 {\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1} is true for all real numbers x {\displaystyle 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

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420-426: 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

455-441: 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. (See the following subsection) If these properties were to define

490-453: Is not restricted to elements of a set. However, many mathematicians refer to both as "Reflexivity", which is generally harmless. 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. 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

525-407: 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 = ? {\displaystyle {\stackrel {?}{=}}} symbol . Viewed as a relation , equality is the archetype of the more general concept of an equivalence relation on

560-403: 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 − 6 x + 5 = 0 {\displaystyle x^{2}-6x+5=0} has

595-410: Is the unique equivalence relation on S {\displaystyle S} whose equivalence classes are all singletons . Given operations over S {\displaystyle S} , that last property makes equality a congruence relation . In logic , a predicate is a proposition which may have some free variables . Equality is a predicate, which may be true for some values of

630-417: 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 ≈ {\displaystyle \approx } ) between real numbers or other things, even if more precisely defined,

665-416: The unit circle in analytic geometry ; therefore, this equation is called the equation of the unit circle . See also: Equation solving 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

700-482: The associated symbol ≅ {\displaystyle \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,

735-417: The axiom of extensionality states that two sets which contain the same elements are the same set. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Lévy. 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

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770-457: 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 . 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,

805-468: The expression " 1 ∪ 2 {\displaystyle 1\cup 2} " (see union ) is an abuse of notation or meaningless. This is a more abstracted framework which can be grounded in ZFC (that is, both axioms can be proved within ZFC as well as most other formal foundations), 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

840-401: The following properties: ∀ a ( a = a ) {\displaystyle \forall a(a=a)} ( a = b ) ⟹ [ ϕ ( a ) ⇒ ϕ ( b ) ] {\displaystyle (a=b)\implies {\bigl [}\phi (a)\Rightarrow \phi (b){\bigr ]}} For example: For all real numbers

875-439: The 💕 [REDACTED] Look up equal , equals , equaling , or equaled in Wiktionary, the free dictionary. Equal ( s ) may refer to: Mathematics [ edit ] Equality (mathematics) . Equals sign (=), a mathematical symbol used to indicate equality. Arts and entertainment [ edit ] Equals (film) , a 2015 American science fiction film Equals (game) ,

910-446: 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 + 2 x + 1 , {\displaystyle (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

945-418: The same object. For example, are two notations for the same number. Similarly, using set builder notation , 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

980-478: 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 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

1015-424: The semantics of expressions and the context. Sometimes, but not always, an identity is written with a triple bar : ( x + 1 ) ( x + 1 ) ≡ x 2 + 2 x + 1. {\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.} In mathematical logic and mathematical philosophy , equality is often described through

1050-565: 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 {\displaystyle 1/2} and 2 / 4 {\displaystyle 2/4} are distinct as fractions (as different strings of symbols) but they "represent"

1085-449: The title Equal . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Equal&oldid=1156977664 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages equal From Misplaced Pages,

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1120-613: The title Equal . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Equal&oldid=1156977664 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Equality (mathematics) A formula such as x = y , {\displaystyle x=y,} where x and y are any expressions, means that x and y denote or represent

1155-624: 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). The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just"). If restricted to the elements of a given set S {\displaystyle S} , those first three properties make equality an equivalence relation on S {\displaystyle S} . In fact, equality

1190-507: The values x = 1 {\displaystyle x=1} and x = 5 {\displaystyle x=5} as its only solutions. The terminology is used similarly for equations with several unknowns. An equation can be used to define a set. For example, the set of all solution pairs ( x , y ) {\displaystyle (x,y)} of the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} forms

1225-423: The variables (if any) and false for other values. More specifically, 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 . See also: Relational operator § Equality An equation

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