Answer :
Final answer:
If Lị is equal to L2, then L3 = (L, NL ) u (Lin L2) is empty, and vice versa.
Explanation:
To prove that Lị is equal to L2 if and only if L3 = (L, NL ) u (Lin L2) is empty, we need to show both directions of the implication.
Direction 1: If Lị is equal to L2, then L3 is empty.
- Assume Lị is equal to L2.
- Since Lị is equal to L2, it means that they recognize the same set of strings.
- Let's consider the language L3 = (L, NL ) u (Lin L2).
- If L3 is not empty, it means there exists a string that is either in L or in NL, or in Lin L2.
- Since Lị is equal to L2, any string in Lin L2 is also in Lị.
- Therefore, if L3 is not empty, it implies that there exists a string in Lị that is either in L or in NL.
- This contradicts the assumption that Lị is equal to L2, as it would mean that there exists a string in Lị that is not in L2.
- Hence, if Lị is equal to L2, then L3 is empty.
Direction 2: If L3 is empty, then Lị is equal to L2.
- Assume L3 is empty.
- Let's consider the language Lị.
- If Lị is not equal to L2, it means they recognize different sets of strings.
- This implies that there exists a string in Lị that is not in L2, or a string in L2 that is not in Lị.
- Since Lị is a regular language, it can be defined using a regular expression or a finite automaton.
- By the definition of L3 = (L, NL ) u (Lin L2), if L3 is empty, it means that there does not exist a string that is either in L or in NL, or in Lin L2.
- Therefore, if L3 is empty, it implies that there does not exist a string in Lị that is either in L or in NL.
- This contradicts the assumption that there exists a string in Lị that is not in L2.
- Hence, if L3 is empty, then Lị is equal to L2.
Therefore, we have proved that Lị is equal to L2 if and only if L3 = (L, NL ) u (Lin L2) is empty.
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