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wolfram-mathematicapiecewise

Check if one of the range conditions is a point

The conditions below yield to a Real point (or hyperplane):

  1. x == 1
  2. x == 1 && y == 2
  3. x == 1 && y < 2
  4. x < 1 && y == 2
  5. x < 1 && y == 2 && z < 5

In other words some of the variables in equations / inequalities above cover only single value rather than range. For obvious reasons equations / inequalities below have all of their variables covering a range and thus the equations themselves represent not a point not a hyperplane but a volume.

  1. x == 1 && y == 2 || (x < 1 && y < 2)
  2. x == 1 && y == 2 || (x < 0 && y < 0)

Is there a way to test if condition belongs to a first or second case (is it a hyperplane or a volume) in Mathematica? I.e. Suppose you have a piecewise function consisting a mixture of conditions above and you want to distinguish conditions based on previously described explanation?

Thank you in advance!

Solution

  • You may use ImplicitRegion with RegionDimension.

    RegionDimension@ImplicitRegion[x == 1 && y == 2 || (x < 1 && y < 2), {x, y}]

    2

    The above is a surface.

    RegionDimension@ImplicitRegion[x == 1 && y == 2, {x, y}]

    0

    The above is a point.

    Hope this helps.