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Suppose we have an inequality $$y>x^2,$$ then obviously the answer is the part of the plane "enclosed" by the parabola.

However, is there any other way to define this part of plane besides "the answer of this inequality" as there are cases when it is not that simple (systems etc.)? If we can precisely define the interval on the number axis, then maybe there is a more formal way in this case? I guess it is possible to extrapolate the question for not only planes, but 3 and more dimensional spaces.

I possess regular highschool mathematics skills, I am just curious.

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    How about $S = \{ (x,y) : y - x^2 > 0 \} \subset \mathbb{R}^2$?2012-04-04
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    Could you please explain this?2012-04-04
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    You consider the 2D plane of points. Each point is described a by a coordinate $(x, y).$ Both $x,y$ are real numbers. So we call this space $\mathbb{R}^2.$ The "part enclosed.. etc" is a subset $S$ of $\mathbb{R}^2.$ This subset is defined as the set of points $(x,y)$ such that $y - x^2 > 0.$ Notation: $S = \{ \text{things} : \text{condition}\}.$2012-04-04
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    Thank you! I suppose if we view discrete points on the plane, then the space would be Z^2, and for higher dimensions we view subsets of space R^n?2012-04-04
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    Correct. $S = \{ (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n : f(x_1, x_2,\ldots, x_n)\mbox{ is true} \}.$ or $\in \mathbb{Z}^n,$ where $f(..)$ is a the condition.2012-04-04
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    Cheers, that was really helpful!2012-04-04

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