(Un)countable set of regular language.

Question

Suppose alphabet Σ={d,e,l,t} and A is the set of all languages "produced" by Σ, which all of them have the property not to include the string "delete". The question is: Is set A countable? I have tried using the Diagonal Argument but it does not work. Any suggestions to a "better" approach?

Answer 1

It is not countable since it is equal to the set of subsets of the set $\Sigma^* - \{delete\}$.

Answer 2

Let $\Sigma_- := \{d,e,l\}$. Then no language over $\Sigma_-$ contains the string "delete" and any language over $\Sigma_-$ is also a language over $\Sigma$. Since the set of all languages over $\Sigma_-$ is uncountable, the claim follows.