# 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

## Examples

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.

## Remarks

In algebraic geometry, solution sets are used to define the Zariski topology. See affine varieties.

## Other meanings

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### Examples

- The solution set for
*E = { x+y = 0 }*w.r.t. is*S = { (a,-a) ; a ∈*.**R**} - The solution set for
*E = { x+y = 0 }*w.r.t. 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 w.r.t. is the interval
*S = [0,2]*(since is undefined for negative values of*x*). - The solution set for w.r.t. is
*S*= 2 π**Z**(see Euler's identity).