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.[1] The symbol "=" is called an "equals sign". Two objects that are not equal are said to be distinct.

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

These last 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 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.

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 asserted to be true for all values of 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.

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.

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).

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

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.

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.[6]

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 the same sets.[8]