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.

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[edit]

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[edit]

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

Other meanings[edit]

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[edit]

  • 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).

See also[edit]