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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. Equality between A and B is written A = B, and pronounced "A equals B". Two objects that are not equal are said to be distinct.
For example:
- means that x and y denote the same object.[1]
- The identity means that if x is any number, then the two expressions have the same value. This may also be interpreted as 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 is 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]
Etymology
[edit]The etymology of the word is from the Latin aequālis ("equal", "like", "comparable", "similar") from aequus ("equal", "level", "fair", "just").[3]
Basic properties
[edit]- 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.
If restricted to the elements of a given set, these three properties make equality an equivalence relation, the unique one whose all equivalence classes are singletons.
- Substitution property: for every a and b, and any assertion (with free variable x), we have that if , then implies [4]
The substitution property has many consequences that include:
- For all real numbers a and b, if a = b, then a ≥ 0 implies b ≥ 0 (here, is x ≥ 0)
- Substitution property for functions: For any a and b, and any function over them, we have that if a = b then (this results from the substitution property applied to ).
- Symmetry and transitivity of equality are implied by reflexivity and the substitution property, cf. Identity and indiscernibility.
Equality as predicate
[edit]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 other values. Equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value (false or true) from its arguments. In computer programming, its computation from the two expressions is known as comparison.
Identities
[edit]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 Sometimes, but not always, an identity is written with a triple bar: [5]
Equations
[edit]An equation is a problem of finding values of some variables, called unknowns, for which the specified equality is true. The term "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, 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: one has to guess an appropriate interpretation from the semantics of expressions and the context. 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.
Approximate equality
[edit]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 symbol.[6]
Relation with equivalence, congruence, and isomorphism
[edit]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.[7] 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.
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 ) 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.[Citation needed]
Equality in set theory
[edit]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
[edit]In first-order logic with equality, the axiom of extensionality states that two sets which contain the same elements are the same set.[8]
- Logic axiom:
- Logic axiom:
- Set theory axiom:
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."[9]
Set equality based on first-order logic without equality
[edit]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.[10]
- Set theory definition:
- Set theory axiom:
See also
[edit]- Extensionality
- Homotopy type theory
- Inequality
- List of mathematical symbols
- Logical equality
- Proportionality (mathematics)
Notes
[edit]- ^ Rosser 2008, p. 163.
- ^ Lévy 2002, pp. 13, 358. Mac Lane & Birkhoff 1999, p. 2. Mendelson 1964, p. 5.
- ^ "Definition of EQUAL". Merriam-Webster. Archived from the original on 15 September 2020. Retrieved 9 August 2020.
- ^ 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
- ^ "Identity – math word definition – Math Open Reference". www.mathopenref.com. Retrieved 1 December 2019.
- ^ https://www.compart.com/en/unicode/U+225F
- ^ (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
References
[edit]- 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.
External links
[edit]- "Equality axioms", Encyclopedia of Mathematics, EMS Press, 2001 [1994]