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I have observed the below statement in a report.

$ f(x,y) = \left\{ \begin{array}{lr} f<0 & : (x,y) \in A\\ f = 0 & : (x,y) \in B \\ f>0 & : (x,y) \in C \end{array} \right.$

I understand the meaning behind it, but is the notation correct? And in this case how do I write the statement then?

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$=$ means "is equal to". So you are saying that $f(x,y)$ is equal to "$f$ is less than $0$" when $(x,y)$ is in $A$, that $f(x,y)$ is equal to "$f$ is equal to $0$" when $(x,y)$ is in $B$, and that $f(x,y)$ is equal to "$f$ is greater than $0$" when $(x,y)\in C$. I would say that one can divine your intent, but it's lousy notation. – Arturo Magidin Jun 18 '12 at 20:08
up vote 1 down vote accepted

It’s at best confusing: I’m not entirely certain what is intended. If the intended meaning is that $f(x,y)$ is negative on $A$, $0$ on $B$, and positive on $C$, you could use the signum function and write

$$\operatorname{sgn}\big(f(x,y)\big)=\begin{cases}-1,&\text{if }(x,y)\in A\\ 0,&\text{if }(x,y)\in B\\ 1,&\text{if }(x,y)\in C\;. \end{cases}$$

And if that’s the intended meaning, I don’t consider what’s in the report correct, though others may be more generous than I.

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No, the notation is not correct, as in its present form it says things like $$f(x,y)=f<0.$$ I would have expressed this more simply as $$f(x,y)<0, \mbox{ for } (x,y)\in A,$$ etc.

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Trying to stay as close as possible to what you quoted, but not to write nonsense, the best I can do is $$ f(x,y)\left\{ \begin{array}{lr} <0 & : (x,y) \in A\\ = 0 & : (x,y) \in B \\ >0 & : (x,y) \in C \end{array} \right. $$ (In "real life" I'd uses LaTeX's "cases" environment, but that doesn't seem to work here.)

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