Solution set
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In mathematics, the solution set of a system of equations or inequality is the set of all its solutions, that is the values that satisfy all equations and inequalities.[1] Also, the solution set or the truth set of a statement or a predicate is the set of all values that satisfy it.
If there is no solution, the solution set is the empty set.[2]
Examples
[edit]- The solution set of the single equation is the set .
- Since there do not exist numbers and making the two equations simultaneously true, the solution set of this system is the empty set .
- The solution set of a constrained optimization problem is its feasible region.
- The truth set of the predicate is .
Remarks
[edit]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
[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 , 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 fi if interpreted as the set of equations fi(x)=0.
Examples
[edit]- The solution set for E = { x y = 0 } with respect to is S = { (a,−a) : a ∈ R }.
- The solution set for E = { x y = 0 } with respect to 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 with respect to is the interval S = [0,2] (since is undefined for negative values of x).
- The solution set for with respect to is S = 2πZ (see Euler's identity).
See also
[edit]References
[edit]- ^ "Definition of SOLUTION SET". www.merriam-webster.com. Retrieved 2024-08-14.
- ^ "Systems of Linear Equations". textbooks.math.gatech.edu. Retrieved 2024-08-14.