Equality (mathematics)

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". Thus there are three kinds of equality, which are formalized in different ways.

Two symbols refer to the same object.^[1]
Two sets have the same elements.^[2]
Two expressions evaluate to the same value, such as a number, vector, function or set.

These may be thought of as the logical, set-theoretic and algebraic concepts of equality respectively.

Etymology

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

Equality in mathematical logic

Logical formulations

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).

In this law, "P(x) if and only if P(y)" can be weakened to "P(x) if P(y)"; the modified law is equivalent to the original.

Instead of considering Leibniz's law as a true statement that can be proven from the usual laws of logic (including axioms about equality such as symmetry, reflexivity and substitution), 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 theorems.

Some basic logical properties of equality

The substitution property states:

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).

In first-order logic, this is a 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 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).

The reflexive property states:

For any quantity a, a = a.

This property is generally used in mathematical proofs as an intermediate step.

The symmetric property states:

For any quantities a and b, if a = b, then b = a.

The transitive property states:

For any quantities a, b, and c, if a = b and b = c, then a = c.

These three properties were originally included among the Peano axioms for natural numbers. Although the symmetric and transitive properties are often seen as fundamental, they can be proved if the substitution and reflexive properties are assumed instead.

Equalities as predicates

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-arguments 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.

Equality in set theory

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 FOL with equality, the axiom of extensionality states that two sets which contain the same elements are the same set.^[3]

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."^[4]

Set equality based on first-order logic without equality

In FOL 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.^[5]

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

Equality in algebra and analysis

Identities

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 (x + 1)² = x² + 2x + 1.

Equations

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, x² + y² = 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.

Congruences

Main articles: Congruence relation and Congruence (geometry)

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 shapes are said equal when one may be moved to coincide with the other. The word congruence is also used for this kind of equality.

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.

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.

Relation with equivalence and isomorphism

Main articles: Equivalence relation and Isomorphism

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 x, consisting of all elements z such that x R z. Then the relation x R y is equivalent with the equality x^R = y^R. 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.^[6] 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

\{\text{A}, \text{B}, \text{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

\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.

However, there are other choices of isomorphism, such as

\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,

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.

Notes

↑ Rosser 2008, p. 163.
↑ Lévy 2002, pp. 13, 358. Mac Lane & Birkhoff 1999, p. 2. Mendelson 1964, p. 5.
↑ 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
↑ (Mazur 2007)

References

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)
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

Hazewinkel, Michiel, ed. (2001), "Equality axioms", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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