binaryoperation
A binary operation on a set S is a rule that combines any two elements a and b in S to produce a third element a*b that also lies in S. Formally it is a function *: S × S → S. The term binary reflects that the operation takes two inputs. In most mathematical contexts the operation is assumed to be total, meaning it is defined for every pair (a, b) in S × S; some discussions allow partial binary operations where the result is undefined for some pairs.
Key properties often discussed for binary operations include closure (the result a*b remains in S), associativity
Examples include addition on the integers, multiplication on the real numbers, string concatenation on the set
Binary operations are foundational in algebra, describing how elements combine and enabling the study of algebraic