Answer :
Final answer:
Operations on context-free languages and regular languages have different properties. Unions and differences between a context-free language and a regular language result in a context-free language, but the intersection of two context-free languages is not necessarily context-free. Therefore the correct answer is option a,b,c.
Explanation:
The question is about analyzing the properties of context-free languages (CFLs) and regular languages with respect to certain operations. A context-free language is generated by a context-free grammar and is recognized by a pushdown automaton.
- L₁ or L₂ is context-free: If L₁ and L₂ are both CFLs, their union (formally written as L₁ ∪ L₂) is also a CFL. This is because the union of two CFLs can be expressed as a context-free grammar by combining the grammars of L₁ and L₂.
- ∪ R is context-free: If L₁ is a CFL and R is a regular language, their intersection (L₁ ∩ R) is a CFL. The operation of intersection with a regular language preserves the context-free property because the result can be constructed using a pushdown automaton for L₁ and a finite automaton for R.
- L₁ - R is context-free: Similarly, if L₁ is a CFL and R is regular, their difference (written as L₁ - R or L₁ \ R) is a CFL as well. This operation can be performed by designing a non-deterministic pushdown automaton that accepts L₁ and rejects R.
- L₁ intersection L₂ is context-free: The intersection of two CFLs L₁ ∩ L₂ is not necessarily a CFL. This is an important exception; while CFLs are closed under union with regular languages, they are not generally closed under intersection with other CFLs.
