Solution set


In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.

For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate to 0), formally

The feasible region of a constrained optimization problem is the solution set of the constraints.


  1. The solution set of the single equation   is the set {0}.
  2. For any non-zero polynomial   over the complex numbers in one variable, the solution set is made up of finitely many points.
  3. However, for a complex polynomial in more than one variable the solution set has no isolated points.



In algebraic geometry, solution sets are called algebraic sets if there are no inequalities. Over the reals, and with inequalities, there are called semialgebraic sets.

Other meanings


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  , supposed to take values in respective spaces  , is the set S of all solutions to the relations E, where a solution   is a family of values   such that substituting   by   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.)

The above meaning is a special case of this one, if the set of polynomials fi if interpreted as the set of equations fi(x)=0.


  • The solution set for E = { x+y = 0 } with respect to   is S = { (a,−a) : aR }.
  • The solution set for E = { x+y = 0 } with respect to   is S = { −y }. (Here, y is not "declared" as an unknown, and thus to be seen as a parameter on which the equation, and therefore the solution set, depends.)
  • The solution set for   with respect to   is the interval S = [0,2] (since   is undefined for negative values of x).
  • The solution set for   with respect to   is S = 2πZ (see Euler's identity).

See also