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3.4.10 Advanced Topic: The Representation of Solution Sets

One can think of any combination of equations or inequalities 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 one just wants 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, one typically needs a more explicit description, of the kind obtained from Reduce.

Here are inequalities that implicitly define a semicircular region.

In[1]:= semi = x > 0 && x^2 + y^2 < 1

Out[1]=

This shows that the point lies in the region.

In[2]:= semi /. { x -> 1/2, y -> 1/2 }

Out[2]=

Reduce gives a more explicit representation of the region.

In[3]:= Reduce[semi, {x, y}]

Out[3]=

If we pick a value for x consistent with the first inequality, we then immediately get an explicit inequality for y.

In[4]:= % /. x -> 1/2

Out[4]=

Reduce[expr, , , ... ] is set up to describe regions by first giving fixed conditions for , then giving conditions for that depend on , then conditions for that depend on and , and so on. This structure has the feature that it allows one to pick points by successively choosing values for each of the in turn—in much the same way as when one uses iterators in functions like Table.

This gives a representation for the region in which one first picks a value for y, then x.

In[5]:= Reduce[semi, {y, x}]

Out[5]=

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 && && ... where each of the is an equation or inequality involving variables up to .

In most cases, however, there will be several components, represented by output containing forms such as || || ... . 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 ... && ( || ) && ... . 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 depend on more of the earlier than is strictly necessary. You can force Reduce to generate results in which a particular only has minimal dependence on earlier 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.

In[6]:= Reduce[x^2 + y == 4 && x^3 - 4y == 8, {x, y}]

Out[6]=

Backsubstituting allows conditions for y to be given without involving x.

In[7]:= Reduce[x^2 + y == 4 && x^3 - 4y == 8, {x, y},
Backsubstitution -> True]

Out[7]=

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.