Let $\Sigma = \{0,1\}$, and suppose that $A$ is a regular language. Define $$A' = \{ u \mid \exists a, b \in\Sigma: abu \in A\}$$ i.e., $A'$ is obtained from $A$ by taking every string in $A$ and removing its first 2 characters. Show that $A'$ is regular.
My solution so far is:
- $ab \in \Sigma^*$ which is regular.
- $A$ is a regular language
- $abu ∈ A$, this means $u$ must be regular for all $abu \in A$ so $A'$ is regular.
Not sure if this makes sense, but please let me know if there's a better way, more formal.
Note: It's not an assignment, I'm just studying for my exam.