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Still at some of my earliest encounters with the elementary concepts of Set Theory, I understand what the Power set of a set is but I have a hard time wrapping my head around what the definition implies in the case of a set like $\Bbb R^2$.

I understand that any function $f:\Bbb R \rightarrow \Bbb R$ is a subset of $\Bbb R^2$

$$\lbrace (x,f(x)): x\in\Bbb R\rbrace \subseteq \Bbb R^2$$

and I can also see why this extends to any set of points in the Cartesian plane

$$\lbrace (x,y): x\in\Bbb R \wedge y \in\Bbb R\rbrace \subseteq \Bbb R^2$$

but then, when I think about what this could mean, my brain starts to hurt a bit. If I can think about something and then write that something down or make a sketch of it, it becomes a collection of points in a plane ergo, by $\lbrace (x,y): x\in\Bbb R \wedge y \in\Bbb R\rbrace \subseteq \Bbb R^2$, it is an element of $\mathscr P(\Bbb R^2)$.

Does that imply any drawing, text, plan, map or even this post could be considered a subset of $\Bbb R^2$ such that $\mathscr P(\Bbb R^2)$ contains any $2$-dimensional "object"?

Since I can make a sketch of most of the things I can think about, does that mean $\mathscr P(\Bbb R^2)$ contains anything that I could think about and sketch and even anything I could sketch but will never think about as well?

Thanks for taking the time to provide your input, I appreciate it a lot!

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    $\begingroup$ Yes, any subset of the plane. A circle, a drawing, any set of points at all. A completely random set of points in the plane is a subset of the plane. The plane contains one dimensional objects too, like lines. $\endgroup$ – user4894 Mar 12 '17 at 18:32
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    $\begingroup$ Yes, any drawing, any two dimensional object you can think about. Even some that are hard to think about, like the set of points with rational coordinates. $\endgroup$ – Ethan Bolker Mar 12 '17 at 18:55
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    $\begingroup$ Also, the full text of your question is in $P(\mathbb R^2)$, as well as the full text of all of the the answers it will receive. $\endgroup$ – Kajelad Mar 12 '17 at 18:57
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    $\begingroup$ Don't even try and think about it like that. There is a reason for layers of abstraction. Use it. $\endgroup$ – mathreadler Mar 12 '17 at 19:31
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Think of it this way: any subset of $\Bbb R ^2$ is any collection of points on a 2D plane. So since a line or a circle can be considered a collection of points in the plane (in fact, that is how they are usually defined), they are in the powerset. Since anything you can sketch can be described as a collection of points, the set of coordinate pairs making up the sketch are in the power set as well.

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