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Relationship asserting that two quantities are the same

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 1=A = B, and pronounced "A equals B". The symbol "1==" is called an "equals sign". Two objects that are not equal are said to be distinct. For example:
means that x and y denote the same object. 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.

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.

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:

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 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. 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.

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.