Multivalued

Multivalued describes a property of a relation, function, or quantity that can take more than one value in a given circumstance. It is used across mathematics, logic, computer science, and information theory to contrast with single-valued objects, which yield exactly one value for each input. Multivalued behavior often arises when a process is nondeterministic, when a mathematical mapping is not a single-valued function, or when the available information is incomplete.

In mathematics, a multivalued function, or multimap, assigns to each input a set of outputs rather than a single output. Formally, F: X → P(Y) maps x in X to a subset F(x) of Y. A familiar example is the square-root relation in the complex numbers: for z ≠ 0, F(z) = {w : w^2 = z} contains two values; for z = 0, F(0) = {0}. To obtain a single value, one may choose a branch or a selection, which yields a conventional single-valued function. Multivalued functions are central in various areas of analysis and topology, where they are treated as relations rather than ordinary functions.

Multivalued logic generalizes classical boolean logic by allowing more than two truth values, such as true, false, and both or neither. This framework models partial or inconsistent information and appears in semantics, reasoning under uncertainty, and certain database theories. In database theory, multivalued attributes describe fields that can contain multiple values for a single record; related concepts include multivalued dependencies and normalization techniques that manage data redundancy and integrity.

Multivaluedness also appears in geometry and analysis through objects like branched coverings and differential inclusions, where a single input may correspond to several possible outputs. In practice, recognizing multivalued relationships helps describe systems with ambiguity, nondeterminism, or incomplete information.