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Answer :
If L1, L1 · L2, and L2 · L1 are all regular, then L2 must also be regular, as regular languages are closed under concatenation is true.
The properties of regular languages under the operations of concatenation imply that if L1 and the concatenations L1 · L2 and L2 ·
L1 are regular, then L2 must also be regular. This is because the regularity of languages is closed under concatenation, meaning the resulting language remains regular if the component languages are regular.
Here's a more detailed explanation:
- If L1 is a regular language, it means there exists a finite automaton that recognizes L1.
- For the concatenations L1 · L2 and L2 · L1 to be regular, there must be finite automata that recognize both concatenated languages.
- Since the automaton for L1 can be combined with those for L2 to form automata for L1 · L2 and L2 · L1, and both concatenated languages are regular, it follows that L2 itself must be regular to maintain this property.
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