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I have a question on if production rules of a formal grammar are being specified correctly. Wikipedia defines the syntax of grammars as the following finite set of production rules, where it states each rule has the form:

$(\Sigma \cup N)^{*}N(\Sigma \cup N)^{*} \implies (\Sigma \cup N)^{*}$

How can each rule have this form when each rule is a subset of the above arguments to the implication symbol? Should quantification and the $\in$ symbol be used to specify each rule? Thanks

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The use of notation in that statement might be confusing. A grammar is a rule of the form "something $\to$ something", where the something on the left is a string of symbols containing at least one non-terminal symbol, and the something on the right is a string of symbols.

For instance, $A \to BcD$ would be a valid rule, as would $AB \to BDeeF$, or $aAcE \to b$.

If you read the very next sentence after the one you quoted, it explains the meaning of the part you didn't understand, in different words. It says "That is, [...]" (you can read the rest yourself). If you don't understand one sentence, in the future I suggest you keep reading a little further to see if you can work it out from context.

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