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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.

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The "painting" $\max\{a,b\}$ is not a statement, but an expression. The value of this expression is undefined if one of $a$ or $b$ (supposedly in ${\mathbb R}$) is undefined. –  Christian Blatter Aug 30 at 14:49

4 Answers 4

up vote 9 down vote accepted

Hint: $$\max\left\lbrace a,b\right\rbrace=\dfrac{a+b+|a-b|}{2}$$

If you don't know $b$ then you can't determine $|a-b|$ and the rest of the formula ;)

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Wonderful answer. Welcome to the community! –  user132181 Aug 30 at 13:38
    
You're very kind, many thanks for the welcome! –  RecentUser Aug 30 at 13:50

The $\text{max}$ operator requires comparison among arguments, or alternatively, requires that the arguments have values which can be used to compute $\operatorname{max}\{a, b\} = \dfrac{a+b+|a-b|}2$. Since we cannot compare an incomparable (since it is undefined) number $b$ with any other number(s), operating on them is rendered meaningless.

It is analogous to asking what $f(x) = x^2$ when evaluated at the undefined value of $b$.

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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.

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If one of the elements is $\infty$ or $-\infty$ (as a limit), then you can extend the definition of $max\{a, b\}$ to be meaningful. I can't think of any other instance where it would make sense to use an argument that you would call undefined.

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