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Assuming $A$ is a set, then $F: A\rightarrow \mathbb{R}$. We can define $F(\varnothing) = 0$. But what is $F(x)$ for $x\notin A$? Is it 0?

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    $\begingroup$ Seriously, if some random person came to you with this very question, would you manage to understand what he's talking about? $\endgroup$ – Saucy O'Path Aug 21 '18 at 17:15
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    $\begingroup$ Given only that $A$ is an arbitrary set, there's no reason to think $\varnothing$ is in $A$, so no reason to think that there is any such thing as $F(\varnothing)$. Also, why are the tags "optimization" and "discrete optimization" there? $\endgroup$ – Malice Vidrine Aug 21 '18 at 17:36
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For any function $F:A \rightarrow B$, the set $A$ is the domain of $F$.

By definition, $F$ assigns elements in $B$ to elements in $A$. In other words, $F(x)$ is not defined for any $x$ which is not in $A$. Or more informally, it has no meaning: you don't use $F$ to evaluate things which are not in its domain.

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