A mathematical expression, function, or construction is well-defined if its defining data determine a unique value or meaning independent of arbitrary choices or representations. Otherwise, it is ill-defined or ambiguous.
For example, the product 
of three integers is well-defined
without parentheses because integer multiplication
is associative, so 
. By contrast, the corresponding operation on Cayley numbers is generally not associative,
so the two parenthesizations can have different values.
Sometimes, ambiguities are implicitly resolved by notational convention. For example, the conventional interpretation of 
is 
, never 
, so that the expression 
is well-defined even though exponentiation is nonassociative.
Well-definedness is distinct from well-posedness. The former concerns whether a definition determines a unique value or meaning; the latter concerns the existence, uniqueness, and stability of solutions to a mathematical problem.
