equalities

In mathematics, an equality is a statement that two expressions denote the same object. It uses the symbol = and is either true or false within a specified domain. Equality has the standard logical properties: reflexivity (a = a), symmetry (if a = b then b = a), and transitivity (if a = b and b = c then a = c). The substitution property lets equal things be replaced by each other in any expression.

Identities are equalities that hold for all admissible values of the variables, such as (x + y)^2 = x^2 + 2xy + y^2. Equations are statements that hold only for particular values, which are sought as solutions. A single equation or a system defines the objects satisfying the conditions.

Solving equalities involves algebraic manipulation to isolate unknowns or determine all satisfying values. Linear equations yield a single or infinite set of solutions; polynomial equations may have several roots; systems are often solved by elimination, substitution, or matrix methods.

Beyond numbers, equalities apply to functions, sequences, and sets. Two functions are equal if they share the same domain and outputs for every input. Sequence equality is termwise; set equality means the same elements. In analysis, exact equalities and equalities that hold almost everywhere can both be meaningful.

In logic and formal systems, equality is a primitive relation governed by axioms such as reflexivity, symmetry, transitivity, and substitution. Care is taken to specify the domain to ensure expressions are defined.