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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 $S = \{ (x,y) : y - x^2 > 0 \} \subset \mathbb{R}^2$? – user2468 Apr 4 '12 at 16:17
Could you please explain this? – end Apr 4 '12 at 16:20
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: $S = \{ \text{things} : \text{condition}\}.$ – user2468 Apr 4 '12 at 16:26
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? – end Apr 4 '12 at 16:30
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. – user2468 Apr 4 '12 at 16:34

Concrete example $$ S = \{ (x,y) : y - x^2 > 0 \} \subset \mathbb{R}^2 $$

In general,

On the real number line, the solution set satisfying a condition $c(x)$ is given by $$ S =\{ x : c(x)\text{ holds} \} \subset \mathbb{R} $$

In $n$-dimensions, the solution for an inequality $c(x_1, x_2, \ldots, x_n)$ is given by the set of points $p = (x_1, x_2, \ldots, x_n) \in \mathbb{R}^n$ such that $c(p)$ holds. Or $$ S =\{ (x_1, x_2, \ldots, x_n) : c(x_1, x_2, \ldots, x_n)\text{ holds} \} \subset \mathbb{R}^n $$ Replace $\mathbb{R}$ by $\mathbb{Z}$ to denote subset of the integer lattice.

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In ${\mathbb R}^n$, a "reasonable" $n$-dimensional set $S$ is typically defined by inequalities of the form $$f_i(x_1,x_2,\ldots, x_n)\geq0\qquad(1\leq i\leq r)\ ,$$ i.e., is defined as $$S:=\bigl\{{\bf x}\in{\mathbb R}^n\ \bigm|\ f_i(x_1,x_2,\ldots, x_n)\geq0\quad (1\leq i\leq r)\bigr\}\ ,\qquad (1)$$ or is a union of sets of this kind. E.g., the full $n$-cube $C$ centered at ${\bf 0}$ with side-length $2$ is described as $$C:=\bigl\{{\bf x}\in{\mathbb R}^n\ \bigm|\ -1\leq x_i\leq 1\quad (1\leq i\leq n)\bigr\}\ ,\qquad(2)$$ and there is no essentially simpler way to describe $C$.

The descriptions $(1)$ and $(2)$ are "implicit". They give easily testable conditions whether an arbitrary given point ${\bf x}=(x_1,,\ldots, x_n)\in{\mathbb R}^n$ belongs to $S$, resp. $C$, or not.

If your set $S$ is more complicated than the cube $C$ you maybe want an explicit "production scheme" that produces each and every point ${\bf x}\in S$, hopefully exactly once, starting from a "standard set" like $C$ or a unit ball $B\subset{\mathbb R}^n$. Such a "production scheme" is called a parametric representation of $S$; it has to be set up ad hoc by means of "mathematical engineering", starting from the equations $(1)$ that define $S$ and using your mathematical expertise about the special functions appearing in $(1)$. In the end you have a hopefully injective map $${\bf g}:\quad C\to{\mathbb R}^n,\quad {\bf u}\mapsto {\bf x}:={\bf g}({\bf u})\ ,$$ that produces for each point ${\bf u}=(u_1,\ldots,u_n)$ in your standard set (say, $C$) a point ${\bf x}=(x_1,\ldots,x_n)\in S$, and it is guaranteed that you obtain all points of $S$ in this way.

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