Regular Expression for $L = \{w \mid w\in \{a,b\}^*\text{ and }n_a(w) \equiv 1 \bmod 3\}$

Question

Here, $Σ=\{a,b\}$ The number of $a$ can be $1, 4, 7, 10.....$, also $a$ can be placed anywhere.

Find Regular Expression for $L = \{w \mid w\in \{a,b\}^*\text{ and }n_a(w) \equiv 1 \bmod 3\}$

How can I proceed in this problem?

Answer 1

You haven't indicated your alphabet, so let me assume that it is $\{a,b\}$, though this makes little difference.

In order to create a regular expression for your language, you can use the following two steps:

1. Write a regular expression for the language of all words $w$ over the alphabet $\{a\}$ such that $n_a(w) \equiv 1 \pmod{3}$.
2. Add back the $b$s to the regular expression. One way to do it is to replace each $a$ by $ab^*$, and to add an additional $b^*$ as a prefix.