Editing Solution set

{{Short description|Set of values which satisfy a given set of equations}}
{{More sources needed|date=January 2011}}

In [[mathematics]], the '''solution set''' of a [[system of equations]] or [[Inequality (mathematics)|inequality]] is the [[set (mathematics)|set]] of all its solutions, that is the values that satisfy all equations and inequalities.<ref>{{Cite web |title=Definition of SOLUTION SET |url=https://www.merriam-webster.com/dictionary/solution+set |access-date=2024-08-14 |website=www.merriam-webster.com |language=en}}</ref> Also, the solution set or the '''truth set''' of a statement or a [[Predicate (mathematical logic)|predicate]] is the set of all values that satisfy it. 

If there is no solution, the solution set is the [[empty set]].<ref>{{Cite web |title=Systems of Linear Equations |url=https://textbooks.math.gatech.edu/ila/systems-of-eqns.html |access-date=2024-08-14 |website=textbooks.math.gatech.edu}}</ref>

== Examples ==
* The solution set of the single equation <math>x=0</math> is the [[singleton set]] <math>\{ 0 \}</math>.
* Since there do not exist numbers <math>x</math> and <math>y</math> making the two equations <math display="block">\begin{cases} x + 2y = 3,&\\   x + 2y = -3 \end{cases}</math> simultaneously true, the solution set of this system is the {{nowrap|[[empty set]] <math>\emptyset</math>.}}
* The solution set of a [[constrained optimization problem]] is its [[feasible region]].
* The truth set of the predicate <math>P(n): n \mathrm{\ is\ even}</math> is <math>\{ 2,4,6,8,\ldots \}</math>.

== Remarks ==
In [[algebraic geometry]], solution sets are called [[algebraic set]]s if there are no inequalities. Over the [[real number|reals]], and with inequalities, there are called [[semialgebraic set]]s.

== Other meanings ==
More generally, the '''solution set''' to an arbitrary collection ''E'' of [[relation (mathematics)|relation]]s (''E<sub>i</sub>'') (''i'' varying in some index set ''I'') for a collection of unknowns <math>{(x_j)}_{j\in J}</math>, supposed to take values in respective spaces <math>{(X_j)}_{j\in J}</math>, is the set ''S'' of all solutions to the relations ''E'', where a solution <math>x^{(k)}</math> is a family of values <math display="inline">{\left( x^{(k)}_j \right)}_{j\in J}\in \prod_{j\in J} X_j</math> such that substituting <math>{\left(x_j\right)}_{j\in J}</math> by <math>x^{(k)}</math> in the collection ''E'' makes all relations "true".

(Instead of relations depending on unknowns, one should speak more correctly of [[Predicate (mathematics)|predicate]]s, 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<sub>i</sub>'' if interpreted as the set of equations ''f<sub>i</sub>''(''x'')=0.

===Examples===

* The solution set for ''E'' = { ''x''+''y'' = 0 } with respect to <math>(x,y)\in \R^2</math> is ''S'' = { (''a'',−''a'') : ''a'' ∈ '''R''' }.
* The solution set for ''E'' = { ''x''+''y'' = 0 } with respect to <math>x \in \R</math> 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 <math> E = \{ \sqrt x \le 4 \} </math> with respect to <math>x\in\R</math> is the interval ''S'' = [0,2] (since <math>\sqrt x</math> is undefined for negative values of ''x'').
* The solution set for <math> E = \{ e^{i x} = 1 \} </math> with respect to <math>x\in\Complex</math> is ''S'' = 2π'''Z''' (see [[Euler's identity]]).

==See also==
* [[Equation solving]]
* [[Extraneous and missing solutions]]

==References==
{{Reflist}}

{{DEFAULTSORT:Solution Set}}
[[Category:Equations]]