Laboratory demand management strategies: An overview

In mathematics, the solution set of a set of equations and inequalities is the set of all its solutions, that is the values that satisfy all equations and inequalities.^[1]

If there is no solution, the solution set is the empty set.^[2]

• The solution set of the single equation is the set {0}.
• Since there do not exist numbers and making the two equations simultaneously true, the solution set of this system is the empty set .
• The solution set of a constrained optimization problem is its feasible region.

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.

More generally, the solution set to an arbitrary collection E of relations (E_i) (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 f_i if interpreted as the set of equations f_i(x)=0.

• The solution set for E = { x+y = 0 } with respect to is S = { (a,−a) : a ∈ R }.
• 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).

• Equation solving
• Extraneous and missing solutions