Question 1Prove that the class of context-free languages is closed under the concatenation operationi.e., if L1𝐿1 and L2𝐿2 are context-free languages, then L1∘L2𝐿1∘𝐿2 is context-free.

To prove that the class of context-free languages is closed under the concatenation operation, we can use the concept of context-free grammars (CFGs).

Step 1: Assume that we have two context-free languages L1 and L2. By definition of context-free languages, there exist two context-free grammars G1 and G2 such that L(G1) = L1 and L(G2) = L2.

Step 2: Let's denote the start symbols of G1 and G2 as S1 and S2 respectively. We can construct a new grammar G as follows:

• The set of nonterminal symbols of G is the union of the set of nonterminal symbols of G1 and G2.
• The set of terminal symbols of G is the union of the set of terminal symbols of G1 and G2.
• The start symbol of G is a new symbol S not used in G1 or G2.
• The set of production rules of G is the union of the set of production rules of G1, G2, and a new rule S → S1S2.

Step 3: It's clear that any string generated by G is a concatenation of a string from L1 and a string from L2, so L(G) is a subset of L1∘L2.

Step 4: Conversely, any string in L1∘L2 can be generated by first applying the rule S → S1S2, then generating a string from L1 using rules from G1 starting from S1, and generating a string from L2 using rules from G2 starting from S2. So, L1∘L2 is a subset of L(G).

Step 5: Since L(G) is a subset of L1∘L2 and L1∘L2 is a subset of L(G), we have L(G) = L1∘L2. Therefore, the concatenation of two context-free languages is also a context-free language. This proves that the class of context-free languages is closed under the concatenation operation.