Solution Set
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High Quality Content by WIKIPEDIA articles! More generally, the solution set to an arbitrary collection E of relations (Ei) (i varying in some index set I) for a collection of unknowns {(x_j)}_{jin J}, supposed to take values in respective spaces {(X_j)}_{jin J}, is the set S of all solutions to the relations E, where a solution x(k) is a family of values {(x^{(k)}_j)}_{jin J}in prod_{jin J} X_j such that substituting {(x_j)}_{jin J} by x(k) in the collection E makes all relations "true". (Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection...
High Quality Content by WIKIPEDIA articles! More generally, the solution set to an arbitrary collection E of relations (Ei) (i varying in some index set I) for a collection of unknowns {(x_j)}_{jin J}, supposed to take values in respective spaces {(X_j)}_{jin J}, is the set S of all solutions to the relations E, where a solution x(k) is a family of values {(x^{(k)}_j)}_{jin J}in prod_{jin J} X_j such that substituting {(x_j)}_{jin J} by x(k) in the collection E makes all relations "true". (Instead of relations depending on unknowns, one should speak more correctly of predicates, the collection E is their logical conjunction, and the solution set is the inverse image of the boolean value true by the associated boolean-valued function.)
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