In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.
Contents
For example, for a set
Examples
1. The solution set of the single equation
2. For any non-zero polynomial
3. However, for a complex polynomial in more than one variable the solution set has no isolated points.
Remarks
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
(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.