TITLE: Is this statement meaningful if one of the elements is undefined?
QUESTION [5 upvotes]: Am I allowed to say a statement like $\max\left\lbrace a,b\right\rbrace$ if it turns out that the element $b$ is undefined, or simply does not exist?  Would the result be $a$, or is the whole statement invalid?
edit: based on the answers I've seen, I think I will have to break my proof into separate cases.  I always want to avoid that when possible, but I would rather make sense in what I write.

REPLY [2 votes]: You're "allowed" to say what you like, as long as you can define what it means.
If it's convenient for your proof to define some quantity that's equal to the maximum of $a$ and $b$ except in the cases where $b$ is undefined, in which case it's equal to $a$, then go right ahead.
The only remaining issue is whether you refer to this quantity as $\max\{a,b\}$, or give it some other name. If you're going to use properties of $\max$ that aren't clearly true of your quantity, then it would be wise to give it a different name. For example using $\max\{a,b\} \ge a$ won't cause any problems. Using $\max\{a,b\} \ge b$ might lead you to make errors, since it's not true (or meaningful even) in the cases for which $b$ is undefined.