Consider the following statements about Context Free Language (CFL):
Statement I: CFL is closed under homomorphism.
Statement II: CFL is closed under complement.
Which of the following is correct?

Correct option is A

1. Statement I:
· CFL (Context-Free Language) is closed under homomorphism.
· A homomorphism is a substitution of symbols in the language's strings with other strings or symbols. CFLs maintain their structure and properties under this operation.
· True: CFL is closed under homomorphism.
2. Statement II:
· CFL is not closed under complement.
· If a language is context-free, its complement may not necessarily be context-free.
· Reason: The complement of a context-free language is not guaranteed to be context-free unless the language is also regular.
· False: CFL is not closed under complement.
Information Booster:
1. Closure Properties of CFLs:
· Closed under:
· Union
· Concatenation
· Kleene star
· Homomorphism
· Reverse
· Not closed under:
· Intersection
· Complement
· Subtraction
2. Key Characteristics:
· CFLs are generated by context-free grammars.
· They are accepted by pushdown automata, which have a stack for memory.
3. Closure under Homomorphism: If you replace each symbol in the language with a string (or another symbol), the resulting language remains context-free.
4. Non-closure under Complement: Closure under complement requires that both the language and its complement be context-free, which is not always true for CFLs.
Additional Knowledge:
· Union of CFLs: CFLs are closed under union because the union of two context-free grammars can be represented as another context-free grammar.
· Intersection of CFLs: CFLs are not closed under intersection, but the intersection of a CFL with a regular language is context-free.
· Pushdown Automata: The inability to handle complement operations stems from the limitations of pushdown automata, as they cannot recognize all context-free complements.