n. 1a. also called essential domain. the set of values of the independent variables of a given function, partial function, or multi-valued function; the set of all the first members of the ordered pairs that constitute the function. In this sense the domain of the real-valued square root function cannot exceed the non-negative reals, and the mapping f of the figure below has the set S' as its domain, as only these members have an image under the mapping.S is the maximal domain and S' the essential domain of f.Compare range.
b. also called maximal domain. the set on which a given function is defined; the set from which the first members of the ordered pairs that formally constitute the function are drawn. In this sense the domain of the real-valued square root function may be taken to be the real numbers, or only the positive reals, or any other set appropriate to the context; for example, the mapping f defined by the diagram above has S as its domain. Compare codomain. 2. an integral domain. See also Euclidean domain. 3. a connected open set. Compare region.