Solution set

As a solution quantity which denotes mathematics , the amount of the solutions of the equation , an inequality , a system of equations and inequalities or general set of (logical) statements.

Solution set

In general, one looks at a set of statements with parameters called variables or unknowns, for example an equation , a system of equations or an inequality . The solution set is the set of assignments of these variables so that all statements of the set are true . Solution sets can be classified according to their size as follows: ${\displaystyle L}$

• ${\displaystyle |L|=0}$: there is no solution (the statements are unsatisfiable; the solution set is empty )
• ${\displaystyle |L|=1}$: there is exactly one solution (the statements are uniquely satisfiable; the solution set consists of exactly one element)
• ${\displaystyle |L|>1}$: there are several, possibly infinitely many, solutions (the statements are satisfiable, but not unique; the solution set consists of more than one element)

The amount of solution also depends on the boundary conditions. For example, the equation for (real numbers) has no solution, but for (complex numbers) it has two solutions. ${\displaystyle x^{2}=-1}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle x\in \mathbb {C} }$

In the case of several solutions, a solution can be specially marked so that a certain clarity is guaranteed. The equation has given for getting two different solutions So , one of which is always a positive and one is negative. Thus the positive (as well as the negative) is unique as such; one defines the root of as the unambiguous positive solution of the given equation. Thus, uniqueness, i.e. the case is enforced by additional boundary conditions (in the example ). However, this is not (meaningfully) possible for all problems. ${\displaystyle x^{2}=a}$ ${\displaystyle a\in \mathbb {R} ^{+}}$ ${\displaystyle x_{1},x_{2}\in \mathbb {R} }$ ${\displaystyle L=\{x_{1},x_{2}\}}$ ${\displaystyle a}$ ${\displaystyle |L|=1}$ ${\displaystyle x>0}$

Solution space

The solution set of a homogeneous or inhomogeneous system of linear equations is always a vector space or an affine space . If the solution set has such a structure, one also speaks of a solution space . If there is an inhomogeneous system of linear equations, i.e. if the mapping matrix of the mapping and a linear mapping between two vector spaces and and is , then there are three possibilities: ${\displaystyle Ax=b}$ ${\displaystyle A}$ ${\displaystyle \Phi \colon V\to W}$ ${\displaystyle \Phi }$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle 0\neq b\in W}$

• The solution set is empty. This is exactly the case when the right side is not in the picture in the figure. ${\displaystyle b}$
• There is exactly one solution , namely if the core of the mapping consists only of the zero vector . ${\displaystyle x}$ ${\displaystyle K}$
• There are infinitely many solutions, whereby all solutions result from any solution by superposition with the solutions of the corresponding homogeneous equation . In this context one calls a particular solution. The solution set is therefore the affine space . ${\displaystyle x_{0}}$ ${\displaystyle Ax=0}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{0}+K}$

Examples

There is an equation and its solution set for : ${\displaystyle x,y\in \mathbb {R} }$

• ${\displaystyle x=1\qquad \;\,L=\left\{1\right\}}$
• ${\displaystyle x+2=5\;\;\,L=\left\{3\right\}}$
• ${\displaystyle {\frac {1}{x^{2}}}=9\qquad L=\{-{\tfrac {1}{3}};{\tfrac {1}{3}}\}}$
• ${\displaystyle x^{2}\leq 4\qquad L=[-2;2]}$, the solution set is an interval
• ${\displaystyle xy=1\qquad L=\left\{\left(x;{\tfrac {1}{x}}\right)\mid x\neq 0\right\}}$, the solution set is a set of pairs .
• A system of linear equations :

${\displaystyle {\begin{matrix}x&+&2y&=&8\\2x&+&y&=&7\\\end{matrix}}\qquad L=\{(2;3)\}}$

literature

• Gerd Fischer : Linear Algebra . 14th edition. Vieweg-Verlag, Wiesbaden 2003, ISBN 3-528-03217-0 .