Order Mistake Definition in CLRS

On page 53 or CLRS it has said :


We can extend our notation to the case of two parameters n and m that can go to infinity independently at different rates. For a given function $g(n,m)$, we denote by $O(g(n,m))$ the set of functions
$$\begin{align}
O(g(n,m)) = \{ f(n,m) : &\text{there exist positive constants }c,
n_0,\text{ and } m_0 \\
&\text{ such that } 0 \le f(n,m) \le cg(n,m)\\
&\text{ for all } n\ge n_0\text{ or } m \ge m_0 \}
\end{align}$$

I think this definition has a problem cause if $n \ge n_0$ for some constant $n$ then we don't need to grow $m$! I feel something is Wrong! It must be the word "and" instead of "or" for example :
$f(x , y ) = x^2 + y^2$

$g_2(x,y) = x^2 + y$

$g_1(x,y) = x^2 + x*y$

by the definition $f(x,y)$ is $O(g_1(x,y))$ !
or even $f(x,y)$ is $O(g_2(x,y))$ ?

please guide me .

Well, there is not really a completely satisfying way to have the big-O Notation with multiple variables. Therefore CLRS present one way of a generalization. Putting an and instead of the or would give a new definition with other problems.

See this article for a discussion on the subject.

StackExchange Computer Science Q#105280, answer score: 2