Solution

The correct answer is A, B, C & D.

Key Points

• Regular expressions (regex) are used to define languages in terms of sets of strings.
• The given options test the properties and interpretations of language operations using regular expressions.
• Each option corresponds to a rule or property of regular expressions and their language sets (L).
• All the given statements (A, B, C, D) are correct as they align with standard interpretations of regex operations.

Step-by-Step Solution / Explanation

• Statement A: L(r₁ + r₂) = L(r₁) ∪ L(r₂)
  • The '+' operator in regex denotes union. Hence, the language represented by r₁ + r₂ is the union of the languages represented by r₁ and r₂.
  • Thus, the statement is correct.
• Statement B: L(r₁ . r₂) = L(r₁)L(r₂)
  • The '.' operator in regex denotes concatenation. Hence, the language represented by r₁ . r₂ is the concatenation of the languages represented by r₁ and r₂.
  • Thus, the statement is correct.
• Statement C: L((r₁)) = L(r₁)
  • Parentheses in regex are used for grouping and do not alter the language represented by the expression.
  • Thus, the language represented by (r₁) is the same as that represented by r₁.
  • Hence, the statement is correct.
• Statement D: L(r₁*) = (L(r₁))*
  • The '*' operator in regex denotes the Kleene star operation, which represents zero or more repetitions of the language.
  • Hence, the language represented by r₁* is the Kleene star of the language represented by r₁.
  • Thus, the statement is correct.

Additional Information

• Regular Expressions Basics:
  • Union (+): Combines the languages of two expressions.
  • Concatenation (.): Combines the strings of two languages in sequence.
  • Kleene Star (*): Represents zero or more repetitions of a language.
• Applications of Regular Expressions:
  • Pattern matching in search engines and text editors.
  • Lexical analysis in compilers.
  • Data validation in input forms.
• Properties of Regex Languages:
  • Union, concatenation, and Kleene star are standard operations in formal language theory.
  • Grouping using parentheses does not alter the language represented.