Solution

The correct answer is A, B and C only

Key Points

• A. φ + L = L + φ = L 
• This expression pertains to the union operation with an empty set φ and a language L.
  1. φ + L: This operation takes the union of an empty set φ with a language set, L. Since there are no elements in the set φ, the union operation with L simply yields the set L. It's basically L union nothing, which is L itself.
  2. L + φ: This is the same scenario as above but the order of sets is reversed. The union operation doesn't depend on the order of sets. It's just L union with nothing, which is still L.
  3. L + φ = φ + L = L simply means that the union of set L with the empty set φ (in either order) gives us L. This is a known property in set theory: the union of any set with the empty set is the set itself.
• B. εL = Lε = L
• This expression pertains to the concatenation operation involving an empty-string ε and a language L.

1. εL: An empty string can be thought of as the "zero" of string operations. Concatenating it in the front of any language L does not alter the language, hence the result is L.
2. Lε: Similarly, concatenating an empty string at the end of any language L also does not change the language, resulting in L.
3. εL = Lε = L simply means that concatenating either at the start or the end of language L with an empty string ε, does not change L.

• C. φL = Lφ = φ
• This expression pertains to the concatenation operation involving an empty set φ and a language L.

1. φL: Concatenating empty set φ in front of (or with) any language L yields an empty set. This is because there are no strings in φ to concatenate with strings in L.
2. Lφ: Similarly, concatenating empty set φ at the end of any language L, also gives empty set, as there are no strings in φ to concatenate with strings in L.
3. φL = Lφ = φ simply means that concatenating either at the start or the end of a language L with an empty set, yields an empty set.

• D. φL = Lφ = L

1. φL and Lφ: As explained above, concatenating an empty set φ (at the start or the end of a language) with any language L will result in an empty set, not L.
2. Therefore, the claim φL = Lφ = L is incorrect. The correct equation would be φL = Lφ = φ.