Final answer:
The true statements are that both L₁ and L₂ are context-free, L₁ and L₂ are both subsets of Σ*, and L₁ ⋅ L₂ is context-free, while it is not necessarily true that L₁ ∪ L₂ is regular, and there is no evidence to support that L₁ is a subset of L₂.
Step-by-step explanation:
The question relates to formal language theory, a topic within the field of computer science and linguistics. When asked about two languages, L₁ and L₂, in this context, we are referring to sets of strings defined by certain rules, not spoken or written human languages. Language L₁ is regular, meaning it can be described by a regular expression or a finite automaton, while L₂ is context-free but not regular, meaning it requires a context-free grammar or a pushdown automaton for its description.
- Both L₁ and L₂ are context free: True. Since regular languages are a subset of context-free languages, L₁ is also context-free by definition.
- L₁ and L₂ are both subsets of Σ*: True. All languages over a given alphabet are subsets of the set of all strings over that alphabet, known as Σ*.
- L₁ ∪ L₂ is regular: False. The union of a regular language and a context-free language is not necessarily regular. It is context-free because the class of context-free languages is closed under union, but it may not be regular if L₂ is not regular.
- L₁ ⋅ L₂ is context-free: True. The concatenation of a regular language and a context-free language results in a context-free language because the class of context-free languages is closed under concatenation.
- L₁ ⊆ L₂: This is False by necessity. There is no information provided that L₁ is a subset of L₂.