Write a grammar that generates the strings over {a,b} starting with a

Question

The answer is:

S -> aA, A -> aA, A -> bA, A -> a, A -> b, S -> a

Any idea how they got this?

Answer 1

Let's write a few strings from the language, some are:$a,aa,abbbb,aabaabbb,...$.

There is only one condition, every string in this language should start with an $a$, we don't care how the following $a$s and $b$s should be. In general the strings in this language are either a single $a$, or an $a$ followed by any sequence of $a$s and $b$s.

Now the rule $S\to a$ makes a single $a$. If something comes after that starting $a$, i.e. some sequence of $a$s and $b$s, then we should use $S \to aA$, Now $A$ produces all non-empty strings with $a$ and $b$ using $A \to aA,A \to bA,A\to a$ or $A\to b$.

Let's see how $ababb$ is made with this grammar: $S \xrightarrow{S\to aA}aA\xrightarrow{A\to bA}abA\xrightarrow{A\to aA}abaA\xrightarrow{A\to bA}ababA\xrightarrow{A\to b}ababb$