Equality (mathematics)

In mathematics, equality is a relationship between two quantities or expressions, stating that they have the same value, or represent the same mathematical object.^[1][2] Equality between A and B is written A = B, and pronounced "A equals B". In this equality, A and B are distinguished by calling them left-hand side (LHS), and right-hand side (RHS).^[3] Two objects that are not equal are said to be distinct.^[4]

Equality is often considered a kind of primitive notion, meaning, its not formally defined, but rather informally said to be "a relation each thing bears to itself and nothing else". This characterization is notably circular ("nothing else"). This makes equality a somewhat slippery idea to pin down.

Basic properties about equality like reflexivity, symmetry, and transitivity have been understood intuitively since at least the ancient Greeks, but weren't symbolically stated as general properties of relations until the late 19th century by Giuseppe Peano. Other properties like substitution and function application weren't formally stated until the development of symbolic logic.

There are generally two ways that equality is formalized in mathematics: through logic or through set theory. In logic, equality is a primitive predicate (a statement that may have free variables) with the reflexive property (called the Law of identity), and the substitution property. From those, one can derive the rest of the properties usually needed for equality. Logic also defines the principle of extensionality, which defines two objects of a certain kind to be equal if they satisfy the same external property (See the example of sets below).

After the foundational crisis in mathematics at the turn of the 20th centrury, set theory (specifically Zermelo–Fraenkel set theory) became the most common foundation of mathematics in order to resolve the crisis. In set theory, any two sets are defined to be equal if they have all the same members. This is called the Axiom of extensionality. Usually set theory is defined within logic, and therefore uses the equality described above, however, if a logic system does not have equality, it is possible to define equality within set theory.