Given the following two grammars : G1 : S → AB | aaB A → a | Aa B → b G2 : S →…

Answer: Both grammars are ambiguous.

Proof for the first grammar (G1):

1. Consider the string "aab".

2. Derivation 1 (using S → AB): S ⇒ AB ⇒ AaB ⇒ aaB ⇒ aab

3. Derivation 2 (using S → aaB): S ⇒ aaB ⇒ aab

4. Since there are two distinct derivations for the same string, G1 is ambiguous.

Proof for the second grammar (G2):

1. Consider the string "abab".

2. Derivation 1 (root expands with a S b S, left S → λ, right S → a S b S with both inner S → λ):

3. S ⇒ a S b S ⇒ a (λ) b (a S b S) ⇒ a b a b

4. Derivation 2 (root expands with a S b S, left S ⇒ b S a S with both inner S ⇒ λ, right S ⇒ λ):

5. S ⇒ a S b S ⇒ a (b S a S) b S ⇒ a (b λ a λ) b λ ⇒ a b a b

6. These are two different parse trees/derivations yielding the same string, so G2 is ambiguous.

Conclusion: Both grammars are ambiguous because we have exhibited specific strings with two distinct derivations in each grammar.