Analyzing Regular Expression Language Properties

This question concerns the fundamental properties defining the languages generated by regular expressions. Let $L(r)$ denote the language (the set of strings) generated by a regular expression $r$. We examine each given property:

Property A: Union Operation

Statement: $L(r_1 + r_2) = L(r_1) \cup L(r_2)$

Explanation: The regular expression operator '+' represents the union. A string belongs to the language $L(r_1 + r_2)$ if it is accepted by either $r_1$ or $r_2$. This directly corresponds to the set-theoretic union of the languages $L(r_1)$ and $L(r_2)$. Therefore, this property is correct.

Property B: Concatenation Operation

Statement: $L(r_1 \cdot r_2) = L(r_1) L(r_2)$

Explanation: The '.' operator represents concatenation. A string belongs to $L(r_1 \cdot r_2)$ if it can be formed by concatenating a string from $L(r_1)$ with a string from $L(r_2)$. This matches the definition of language concatenation. Therefore, this property is correct.

Property C: Parentheses Grouping

Statement: $L((r_1)) = L(r_1)$

Explanation: Parentheses in regular expressions are used for grouping sub-expressions to control the order of operations. They do not alter the set of strings accepted by the expression. Thus, the language generated by $(r_1)$ is identical to that generated by $r_1$. This property is correct.

Property D: Kleene Star Operation

Statement: $L(r_1^*) = (L(r_1))^*$

Explanation: The Kleene star operator ($*$) applied to a regular expression $r_1$ generates all possible sequences of zero or more concatenations of strings belonging to the language $L(r_1)$. This is precisely the definition of the Kleene star applied to a language. Therefore, this property is correct.

Conclusion

All four statements (A, B, C, and D) represent standard and correct properties relating regular expressions to the languages they generate.