Solution set
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. 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.
Examples
- The solution set of the single equation is the singleton set.
- Since there do not exist numbers and making the two equations simultaneously true, the solution set of this system is the
- The solution set of a constrained optimization problem is its feasible region.
- The truth set of the predicate is.
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 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".The above meaning is a special case of this one, if the set of polynomials fi if interpreted as the set of equations fi=0.
Examples
- The solution set for E = with respect to is S =.
- The solution set for E = with respect to is S =.
- The solution set for with respect to is the interval S = .
- The solution set for with respect to is S = 2πZ.