 # Solution set

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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 { 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 R : i 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 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 ( x j ) j J , supposed to take values in respective spaces ( X j ) j J , is the set S of all solutions to the relations E, where a solution x ( k ) is a family of values ( x j ( k ) ) j J j J X j such that substituting ( x j ) j 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 ) R 2 is S = { (a,-a) ; a ∈ R } .
• The solution set for E = { x+y = 0 } w.r.t. x 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 = { x 4 } w.r.t. x R is the interval S = [0,2] (since x is undefined for negative values of x).
• The solution set for E = { exp ( i x ) = 1 } w.r.t. x C is S = 2 π Z (see Euler's identity).
• ## References

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