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 ${\displaystyle x=0}$ is the singleton set ${\displaystyle \{0\$ .
Since there do not exist numbers ${\displaystyle x}$ and ${\displaystyle y}$ making the two equations ${\displaystyle {\begin{cases}x+2y=3,&\\x+2y=-3\end{cases$ simultaneously true, the solution set of this system is the empty set ${\displaystyle \emptyset }$ .
The solution set of a constrained optimization problem is its feasible region.
The truth set of the predicate ${\displaystyle P(n):n\mathrm {\ is\ even} }$ is ${\displaystyle \{2,4,6,8,\ldots \$ .

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 (E_i) (i varying in some index set I) for a collection of unknowns ${\displaystyle {(x_{j})}_{j\in J$ , supposed to take values in respective spaces ${\displaystyle {(X_{j})}_{j\in J$ , is the set S of all solutions to the relations E, where a solution ${\displaystyle x^{(k)$ is a family of values ${\textstyle {\left(x_{j}^{(k)}\right)}_{j\in J}\in \prod _{j\in J}X_{j$ such that substituting ${\displaystyle {\left(x_{j}\right)}_{j\in J$ by ${\displaystyle x^{(k)$ 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 f_i if interpreted as the set of equations f_i(x)=0.

Examples#edit

The solution set for E = { x+y = 0 } with respect to ${\displaystyle (x,y)\in \mathbb {R} ^{2$ is S = { (a,−a) : a ∈ R }.
The solution set for E = { x+y = 0 } with respect to ${\displaystyle x\in \mathbb {R} }$ 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 ${\displaystyle E=\\sqrt {x\leq 4\$ with respect to ${\displaystyle x\in \mathbb {R} }$ is the interval S = [0,16] (since ${\displaystyle {\sqrt {x$ is undefined for negative values of x).
The solution set for ${\displaystyle E=\{e^{ix}=1\$ with respect to ${\displaystyle x\in \mathbb {C} }$ is S = 2πZ (see Euler's identity).

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.

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Solution set

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