Which of the following are correct on regular expressions? A.…

Evaluate each given statement about languages:

• Statement: φ + L = L + φ = L (union with the empty set) - True. The empty set contains no strings, so taking the union with a language L adds nothing; therefore φ ∪ L = L.

• Statement: εL = Lε = L (concatenation with the empty string) - True. Concatenating any string s with the empty string ε yields s, so {ε}·L = L and L·{ε} = L.

• Statement: φL = Lφ = φ (concatenation with the empty set) - True. Since φ has no strings, there are no pairs to concatenate, so the concatenation is empty: φ·L = ∅ and L·φ = ∅.

• Statement: φL = Lφ = L (concatenation with the empty set yields the original language) - False. This contradicts the previous correct statement: concatenation with the empty set cannot produce strings, so it does not yield L.

Conclusion: The true statements are the ones asserting that union with the empty set leaves a language unchanged, concatenation with the empty string leaves a language unchanged, and concatenation with the empty set yields the empty set. Any choice that includes the claim that concatenating with the empty set yields the original language is incorrect.