|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{Use dmy dates|date=July 2013}}
| | Hi there. Allow me begin by introducing the author, her name is Myrtle Cleary. Years ago we moved to North Dakota. I am a meter reader but I plan on altering it. What I love performing is to gather badges but I've been taking on new issues lately.<br><br>my blog ... diet meal delivery ([http://remedies-and-cure.info/dietdelivery10007 have a peek at this web-site]) |
| In [[mathematics]] '''equality''' is a relationship between two quantities or, more generally two [[mathematical expression]]s, 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]]".
| |
| | |
| When ''A'' and ''B'' may be viewed as [[function (mathematics)|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 (mathematics)|identity]]. An example is (''x'' + 1)<sup>2</sup> = ''x''<sup>2</sup> + 2''x'' + 1.
| |
| | |
| When ''A'' and ''B'' are not fully specified or depend on some [[Variable (mathematics)|variables]], equality is a [[proposition (mathematics)|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-arguments [[predicate (mathematical logic)|predicate]] which may produce a [[truth value]] (''false'' or ''true'') from its arguments. In [[computer programming]], its computation from two expressions is known as [[relational operator|comparison]].
| |
| | |
| In some cases, one may consider as '''equal''' two mathematical objects that are only equivalent for the properties that are considered. This is, in particular the case in [[geometry]], where two [[geometric shape]]s are said equal when one may be moved to coincide with the other. The word '''congruence''' is also used for this kind of equality.
| |
| | |
| An [[equation]] is the 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 on. For example ''x''<sup>2</sup> + ''y''<sup>2</sup> = 1 is the ''equation'' of the [[unit circle]]. There is no standard notation which distinguishes an equation from an identity or other use of the equality relation: a reader has to guess an appropriate interpretation from the semantic of expressions and the context.
| |
| | |
| There are several formalizations of the notion of equality in [[mathematical logic]], usually by means of axioms, such as the first few [[Peano axioms]], or the [[axiom of extensionality]] in [[Zermelo–Fraenkel set theory|ZF set theory]]). There are also some [[mathematical logic|logic systems]] that do not have any notion of equality. This reflects the [[undecidable problem|undecidability]] of the equality of two [[real number]]s defined by formulas involving the [[integer]]s, the basic [[arithmetic operation]]s, the [[logarithm]] and the [[exponential function]]. In other words,
| |
| there cannot exist any [[algorithm]] for deciding such an equality.
| |
| | |
| Viewed as a relation, equality is the archetype of the more general concept of an [[equivalence relation]] on a set: those binary relations which are [[reflexive relation|reflexive]], [[symmetric relation|symmetric]], and [[transitive relation|transitive]].
| |
| The identity relation is an equivalence relation. Conversely, let ''R'' be is an equivalence relation, and let us denote by ''x<sup>R</sup>'' 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<sup>R</sup>'' = ''y<sup>R</sup>''. It follows that equality is the smallest 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).
| |
| | |
| The etymology of the word is from the Latin [[wikt:aequalis|''aequalis'']], meaning uniform or identical, from ''aequus'', meaning "level, even, or just."
| |
| | |
| ==Logical formulations==
| |
| Equality is always defined such that things that are equal have all and only the same properties. Some people{{who|date=January 2014}} define equality as congruence. Often equality is just defined as [[Identity (philosophy)|identity]].
| |
| | |
| A stronger sense of equality is obtained if some form of [[Identity of indiscernibles|Leibniz's law]] is added as an [[axiom]]; the assertion of this axiom rules out "bare particulars"—things that have all and only the same properties but are not equal to each other—which are possible in some logical formalisms. The axiom states that two things are equal if they have all and only the same [[Property (philosophy)|properties]]. Formally:
| |
| : [[Given any]] ''x'' and ''y'', ''x'' = ''y'' [[material conditional|if]], given any [[Predicate (mathematics)|predicate]] ''P'', ''P''(''x'') [[if and only if]] ''P''(''y'').
| |
| | |
| In this law, the connective "if and only if" can be weakened to "if"; the modified law is equivalent to the original.
| |
| | |
| Instead of considering Leibniz's law as an axiom, it can also be taken as the ''definition'' of equality. The property of being an equivalence relation, as well as the properties given below, can then be proved: they become [[theorem]]s.
| |
| If a=b, then a can replace b and b can replace a.
| |
| | |
| ==Some basic logical properties of equality==
| |
| The substitution property states:
| |
| * [[For any]] quantities ''a'' and ''b'' and any expression ''F''(''x''), [[material conditional|if]] ''a'' = ''b'', then ''F''(''a'') = ''F''(''b'') (if either side makes sense, i.e. is [[well-formed formula|well-formed]]).
| |
| In [[first-order logic]], this is a [[schema (logic)|schema]], since we can't quantify over expressions like ''F'' (which would be a [[functional predicate]]).
| |
| | |
| Some specific examples of this are:
| |
| * For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''a'' + ''c'' = ''b'' + ''c'' (here ''F''(''x'') is ''x'' + ''c'');
| |
| * For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''a'' − ''c'' = ''b'' − ''c'' (here ''F''(''x'') is ''x'' − ''c'');
| |
| * For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'', then ''ac'' = ''bc'' (here ''F''(''x'') is ''xc'');
| |
| * For any [[real number]]s ''a'', ''b'', and ''c'', if ''a'' = ''b'' and ''c'' [[Division by zero|is not]] [[0 (number)|zero]], then ''a''/''c'' = ''b''/''c'' (here ''F''(''x'') is ''x''/''c'').
| |
| | |
| The reflexive property states:
| |
| :[[For any]] quantity ''a'', ''a'' = ''a''.
| |
| | |
| This property is generally used in [[mathematical proof]]s as an intermediate step.
| |
| | |
| The symmetric property states:
| |
| * [[For any]] quantities ''a'' and ''b'', [[material conditional|if]] ''a'' = ''b'', then ''b'' = ''a''.
| |
| | |
| The transitive property states:
| |
| * [[For any]] quantities ''a'', ''b'', and ''c'', [[material conditional|if]] ''a'' = ''b'' [[and (logic)|and]] ''b'' = ''c'', then ''a'' = ''c''.
| |
| | |
| The [[binary relation]] "[[approximation|is approximately equal]]" between [[real number]]s or other things, even if more precisely defined, is not transitive (it may seem so at first sight, but many small [[Difference (mathematics)|differences]] can add up to something big).
| |
| However, equality [[almost everywhere]] ''is'' transitive.
| |
| | |
| Although the symmetric and transitive properties are often seen as fundamental, they can be proved, if the substitution and reflexive properties are assumed instead.
| |
| | |
| ==Relation with equivalence and isomorphism==
| |
| {{See also|Equivalence relation|Isomorphism}}
| |
| | |
| In some contexts, equality is sharply distinguished from ''[[equivalence relation|equivalence]]'' or ''[[isomorphism]].''<ref>{{Harv|Mazur|2007}}</ref> For example, one may distinguish ''[[fraction (mathematics)|fractions]]'' from ''[[rational number]]s,'' the latter being equivalence classes of fractions: the fractions <math>1/2</math> and <math>2/4</math> 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
| |
| :<math>\{\text{A}, \text{B}, \text{C}\} \,</math> and <math>\{ 1, 2, 3 \} \,</math>
| |
| | |
| 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
| |
| :<math>\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.</math>
| |
| | |
| However, there are other choices of isomorphism, such as
| |
| :<math>\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,</math>
| |
| | |
| and these sets cannot be identified without making such a choice – any statement that identifies them "depends on choice of identification". This distinction, [[Isomorphism#Relation_with_equality|between equality and isomorphism]], is of fundamental importance in [[category theory]], and is one motivation for the development of category theory.
| |
| | |
| ==See also==
| |
| *[[Equals sign]]
| |
| *[[Inequality (mathematics)|Inequality]]
| |
| *[[Logical equality]]
| |
| *[[Extensionality]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| {{Refbegin}}
| |
| * {{Citation | first = Barry | last = Mazur | authorlink = Barry Mazur | title = When is one thing equal to some other thing? | date = 12 June 2007 | url = http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf }}
| |
| * {{Cite book
| |
| | authorlink = Saunders Mac Lane
| |
| | first = Saunders
| |
| | last = Mac Lane
| |
| | coauthors = [[Garrett Birkhoff]]
| |
| | title = Algebra
| |
| | publisher = American Mathematical Society
| |
| | year = 1967}}
| |
| {{Refend}}
| |
| | |
| {{DEFAULTSORT:Equality (Mathematics)}}
| |
| [[Category:Mathematical logic]]
| |
| [[Category:Mathematical relations]]
| |
| [[Category:Elementary arithmetic]]
| |