# 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 $\{f_i\}$ of polynomials over a ring $R$, the solution set is the subset of $R$ on which the polynomials all vanish (evaluate to 0), formally

$\{x\in R:\forall i\in I, f_i(x)=0\}.\$

## Examples

1. The solution set of the single equation $x=0$ is the set  {0}.

2. For any non-zero polynomial $f$ 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 (Ei) (i varying in some index set I) for a collection of unknowns ${(x_j)}_{j\in J}$, supposed to take values in respective spaces ${(X_j)}_{j\in J}$, is the set S of all solutions to the relations E, where a solution $x^{(k)}$ is a family of values ${(x^{(k)}_j)}_{j\in J}\in \prod_{j\in J} X_j$ such that substituting ${(x_j)}_{j\in J}$ 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.)

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.

### Examples

• The solution set for E = { x+y = 0 } w.r.t. $(x,y)\in\mathbb R^2$ is S = { (a,-a) ; a ∈ R } .
• The solution set for E = { x+y = 0 } w.r.t. $x\in\mathbb R$ 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 $E = \{ \sqrt x \le 4 \}$ w.r.t. $x\in\mathbb R$ is the interval S = [0,2] (since $\sqrt x$ is undefined for negative values of x).
• The solution set for $E = \{ \exp(i x) = 1 \}$ w.r.t. $x\in\mathbb C$ is S = 2 π Z (see Euler's identity).