# Is this the right notation for points in a square or rectangle?

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 $\{(x, y) \in [-10; 10]\times [-20; 5] \}$?

$$H = \{(x,y): (x,y) \in [-10,10] \times [-10,10]\}$$

• Would you say the shorthand notation of $^2$ is invalid if the intervals are the same? – null Aug 10 '16 at 16:23
• Nah, definitely valid. – Faraad Armwood Aug 10 '16 at 16:24
• $[-10,10]^2$ is valid. It might be ambiguous or unclear in another context. But it is valid. However H = {$z|z \in$ G} is redundant and pointlessly over nested. – fleablood Aug 10 '16 at 16:37

$[a,b] \times [c,d]$, 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.

• Things like this are a matter of opinion. I didn't know people cared so much. I can delete my response if it's really that repulsive. – Faraad Armwood Aug 10 '16 at 16:42
• "repulsive" is too strong an overstatement. it was in response to the OP not you.you were the accepted answer. Please do not delete. [a,b] x [c,d] is acceptable notation as we point out. My point is if you use set notation then use it. i)H = {x| x in G} means H=G. Always. so don't pussyfoot and ii)H={some set} form is for the sake of definition. If you use set notation you can use it to define things; do so. "What is the notation for the set of all real numbers whose cube root has third decimal digit = 7?" Answer: H = {x|cube root of x has 7 as its third decial digit} – fleablood Aug 10 '16 at 16:52
• I just think at all times we should remain as professional as possible on this site. I understand what you are saying now and sorry for the confusion. – Faraad Armwood Aug 10 '16 at 16:54
• "why the heck don't you your just write H=[a,b]×[c,d]H=[a,b]×[c,d]?" because I don't need to give the set a name, but I do want to use the variables x and y, because they are used previously. I never liked the notation with inequalities and prefer the interval notation. I just want to say in what area I made the plot. – null Aug 10 '16 at 17:30
• @null point taken but that's kind of my point. If x, and why were used explicitly then they are not the generic (x,y) in {(x,y)|(x,y) \in [a,b][,cd]}. Just say "I plotted all the (x,y) in [a,b]x[c,d]". To say "I plotted all the (x,y) such that (x,y) were in the set of all (w,y) where (w,y) were in [a,b]x[c,d]" is pointlessly nested. You could say "I plotted all the points in {(x,y)| x in [a,b] and y in [c,d]}" if you don't like inequalities but I think you can refer to notation "I plotted all the points in [a,b]x[c,d]" and and assume it will be understood. – fleablood Aug 10 '16 at 17:43