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I'm trying to write down the definition of the type-3 grammar in pure mathematics and there is a rule $S \rightarrow \epsilon$ which can be in the ruleset under a certain condition. I've come up with this notation: $$A=B\cup \begin{cases} C & \nexists x \in A : P(x) \\ \emptyset & otherwise \end{cases}\quad.$$ Of course, the claim $\forall x \in C: \neg P(x)$ is valid, but I don't know if the aforementioned notation is right and I don't like it anyway so I'm asking for a better (more compact) one.

EDIT:

The more readable version: $$A=B\cup \begin{cases} C & \nexists x \in B : P(x) \\ \emptyset & otherwise \end{cases}$$

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  • $\begingroup$ If you want to define $A$ by this notation, then this notation suggests that the definition of $A$ depends on what $A$ is, which is not a good idea. Do you mean "there is no $x$ in $B$" in the first line of your alternative? $\endgroup$
    – Goldstern
    Aug 22, 2013 at 13:45
  • $\begingroup$ Actually, yes. It depends on what is in $B$ and not $C$, but with the supplemented proposition it should be ok. Nonetheless I'm going to do an edit. $\endgroup$
    – Goheeca
    Aug 22, 2013 at 16:28

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