Solution set

In mathematics, the set of solutions of an equation, an inequality, a system of equations and inequalities or general set of ( logical) statements is often called the solution set.

Amount of solution

Generally speaking you have a lot of statements with parameters, variables or unknowns are called, for example, an equation, a system of equations or inequality. As a solution set is now denotes the set of assignments of these variables, so that all statements are true of the crowd. Can be classified according to their size solution sets as follows:

  • : There is no solution ( the statements are unsatisfiable and the solution set is empty )
  • : There is exactly one solution ( the statements are clearly satisfiable, the solution set consists of exactly one element )
  • : There are several, potentially infinite number of solutions ( the statements are satisfiable, but not unique, the solution set consists of more than one element )

The amount of solution depends on both the set of propositions themselves, as well as the domain of the variables. For example, the equation for ( real numbers ) has no solution, while for ( complex numbers) two solutions.

Solution space

The solution set of a homogeneous or inhomogeneous linear system of equations is always a vector space, or an affine space. Does the solution set such a structure, one also speaks of a solution space. Is a non-homogeneous linear system of equations, so the projection matrix of the image and a linear map between two vector spaces and and, then there are three possibilities:

  • The solution set is empty. This is exactly the case when the right side is not in the image of the figure.
  • There is exactly one solution, namely, when the core of the image consists of only the zero vector.
  • There are infinitely many solutions, where all solutions of an arbitrary solution by superposition result with the solutions of the corresponding homogeneous equation. It is called in this context a particular solution. The solution set is thus the affine space.

Examples

There is one equation and its solutions for specified:

  • , The solution set is an interval
  • , The solution set is a set of pairs.
  • A linear system of equations:
Uniqueness quantification Transformation matrix Null vector
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