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I want to express through a variable, that a certain action is either done, or not. When the action is done, the variable should have value 1, if not, the variable should be empty. How do I express the scope of this variable?

$\in \left\{0, 1\right\}$ doesn't feel quite right, maybe $\in \emptyset, \left\{1\right\}$?

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    @Asaf: I understood this to be shorthand for saying "what do I need to put after the variable to express this property of the variable".2011-05-12

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Say in words "a value of zero corresponds to no action". If the value zero corresponds to a feasible action, then use something else that is outside the action space. I see this in papers on decision theory.

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You can say that if $x$ is your variable then $|x|\le 1$ and $|x| > 0$ if and only if the action was done.