I have a function $f$ that calculates a vector for a given point, both in 2D (that's to say $f:\mathbb R^2 \to \mathbb R^2, (x,y) \mapsto (u,v)$, I guess).
I plotted the resulting field of vectors limiting both x and y to the interval $[-10, 10]$, thus giving me a square around the origin. How do I put this in the set notation? I guess I could say I plotted in $\{(x, y) \in [-10; 10]^2\}$. Is that correct?
What if one dimension used a different interval (creating a rectangle)? What would be the right symbol between both intervals? Could I write $\{(x, y) \in [-10; 10]\times [-20; 5] \}$?
Best Answer
$[a,b] \times [c,d]$, I think, would be recognized as the set you mean.
In my opinion, writing $H = \{(x,y)| (x,y) \in [a,b] \times [c,d]\}$ is redundant. If $(x,y)$ is in $[a,b] \times [c,d]$ then why the heck don't you your just write $H = [a,b]\times [c,d]$????? After all if you wrote $H = \{x|x \in \mathbb R\}$ that'd be seen as ridiculous; $H = \mathbb R$ fercripesake!
The only reason to put it in set notation is for clarity/definition as $[a,b]\times [c,d]$ might not be clear in meaning or might not be known to a novice. In which case $H = \{(x,y)| (x,y) \in [a,b] \times [c,d]\}$ does nothing to add to the clarity.
Why not just write $H = \{(x,y)| a \le x \le b; c \le y \le d\}$. That's perfectly clear and legit. Or $H = [a,b]\times [c,d] = \{(x,y)| a \le x \le b; c \le y \le d\}$ can be seen as a definition.
But to answer your question $H = [a,b]\times[c,d] := \{(x,y)| a \le x \le b; c \le y \le d\}$ is acceptable and standard notation.