Creating a language

Question

I am given a list languages, say $L$, over alphabet $\{a,b\}$. A function $f$ is defined such that $f(i) = L$ for $i ∈ N$. I am trying to a construct a language $D$ which is not in the list (aka. $D \neq f(n)$). I am not quite sure how to construct a language, I started off by constructing a language of $i$ that is not a natural number such as $N = \{-2, -1\}$ but I am not quite if this is correct? What can be concluded from that language?

Answer 1

Hint: Consider the language $D$ defined by:

1. $D$ contains no word that has a $\mathtt b$ in it.

2. For every $n\in\mathbb N$, the word $\mathtt a^n$ (that is, $\underbrace{\mathtt{aaa\ldots a}}_{n\text{ a's}}$) is in $D$ if and only if it is not in $f(n)$.

Can you prove that $D$ is not in the list?