Use set notation to define the language generated by the grammar

[EonsNearby]

Homework Statement

As the topic states, I need to define a language, for a grammar, using set notation.

Homework Equations

Here is the grammar:
S -> aaSB | λ
B -> bB | b

The Attempt at a Solution

Okay, I know that this creates strings that are either empty or consist of an even number of 'a' characters followed by 1 or more 'b' characters.

My first set notation is as follows:
{(a^2m)(b^np) | m ≥ 0, n > 0, 0 < p ≤ m}
Another variation of this I thought of but discarded (b/c I thought it would give negative exponents) is as follows:
{(a^2m)(b^np) | m ≥ 0, n > 0, p ≤ m}

This is a separate set notation I came up with (but I'm less confident about it):
{(a^2m)(b^n) | m > 0, n > 0} U {(a^m)(b^m) | m = 0}

Are either of these correct, and if not, could someone help me come up with a correct one?

 

[KataKoniK]

Your first set notation is almost correct, but there is one small error. The correct notation for the grammar would be:

{(a^2m)(b^np) | m ≥ 0, n > 0, p ≥ 0, p ≤ m}

This is because the grammar allows for the string "bb" to be generated, which would not be included in your notation. The addition of "p ≥ 0" ensures that the grammar can generate any number of "b" characters, including 0.

Your second set notation is not correct, as it does not account for the possibility of an empty string.

Your third set notation is also not correct, as it does not account for the possibility of an even number of "a" characters followed by an odd number of "b" characters.

So, the correct set notation for the grammar would be:

{(a^2m)(b^np) | m ≥ 0, n > 0, p ≥ 0, p ≤ m} U {(a^2m)(b^np) | m ≥ 0, n > 0, p = 0}