The Representation of Solution Sets

Any combination of equations or inequalities can be thought of as implicitly defining a region in some kind of space. The fundamental function of Reduce is to turn this type of implicit description into an explicit one.

An implicit description in terms of equations or inequalities is sufficient if you just want to test whether a point specified by values of variables is in the region. But to understand the structure of the region, or to generate points in it, you typically needs a more explicit description, of the kind obtained from Reduce.

Here are inequalities that implicitly define a semicircular region:
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This shows that the point lies in the region:
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Reduce gives a more explicit representation of the region:
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If you pick a value for x consistent with the first inequality, you then immediately get an explicit inequality for y:
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Reduce[expr,{x1,x2,}] is set up to describe regions by first giving fixed conditions for x1, then giving conditions for x2 that depend on x1, then conditions for x3 that depend on x1 and x2, and so on. This structure has the feature that it allows you to pick points by successively choosing values for each of the xi in turnin much the same way as when you uses iterators in functions like Table.

This gives a representation for the region in which you first pick a value for y, then x:
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In some simple cases the region defined by a system of equations or inequalities will end up having only one component. In such cases, the output from Reduce will be of the form e1&&e2, where each of the ei is an equation or inequality involving variables up to xi.

In most cases, however, there will be several components, represented by output containing forms such as u1||u2||. Reduce typically tries to minimize the number of components used in describing a region. But in some cases multiple parametrizations may be needed to cover a single connected component, and each one of these will appear as a separate component in the output from Reduce.

In representing solution sets, it is common to find that several components can be described together by using forms such as &&(u1||u2)&&. Reduce by default does this so as to return its results as compactly as possible. You can use LogicalExpand to generate an expanded form in which each component appears separately.

In generating the most compact results, Reduce sometimes ends up making conditions on later variables xi depend on more of the earlier xi than is strictly necessary. You can force Reduce to generate results in which a particular xi only has minimal dependence on earlier xi by giving the option Backsubstitution->True. Usually this will lead to much larger output, although sometimes it may be easier to interpret.

By default, Reduce expresses the condition on y in terms of x:
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Backsubstituting allows conditions for y to be given without involving x:
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generate the cylindrical algebraic decomposition of the region defined by expr
find the full-dimensional part of the decomposition of the region defined by expr, together with any hypersurfaces containing the rest of the region
give at least one point in each connected component of the region defined by expr

Cylindrical algebraic decomposition.

For polynomial equations or inequalities over the reals, the structure of the result returned by Reduce is typically a cylindrical algebraic decomposition or CAD. Sometimes Reduce can yield a simpler form. But in all cases you can get the complete CAD by using CylindricalDecomposition. For systems containing inequalities only, GenericCylindricalDecomposition gives you "most" of the solution set and is often faster.

Here is the cylindrical algebraic decomposition of a region bounded by a hyperbola:
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This gives the two-dimensional part of the solution set along with a curve containing the boundary:
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This finds solutions from both parts of the solution set:
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The results include a few points from each piece of the solution set:
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