Use the string “00111” to show that the following grammar is ambiguous: 𝑆 → 𝐴𝑆 | ϵ 𝐴 → 𝐴1 | 0𝐴1 | 01

The given string is "00111". The grammar is ambiguous if there is more than one derivation tree that leads to the same string. Let's see if we can derive the string in two different ways.

First derivation:

1. S -> AS (using S -> AS)
2. AS -> 0A1S (using A -> 0A1)
3. 0A1S -> 00A11S (using A -> 0A1)
4. 00A11S -> 00111S (using A -> 01)
5. 00111S -> 00111 (using S -> ϵ)

Second derivation:

1. S -> AS (using S -> AS)
2. AS -> 0A1S (using A -> 0A1)
3. 0A1S -> 01S (using A -> 01)
4. 01S -> 011S (using S -> AS and A -> 01)
5. 011S -> 0111S (using S -> AS and A -> 01)
6. 0111S -> 00111 (using S -> ϵ)

As we can see, there are two different derivations for the same string "00111". Therefore, the given grammar is ambiguous.