Is the empty set/language contained in the following set

Question

Assume I have the following set of languages: $$ \{L \subseteq \{0,1\}^* \mid \text{for all $w \in L$, $|w| \leqslant 3$}\} $$ I know it contains the language containing the empty word since the empty word's length is $0$, but does it also contain the empty set\language ?

Answer 1

Let's go through the question in detail. First, if $\emptyset$ is in the set of languages, then we need $\emptyset\subseteq \{0,1\}^*$. This is true because the empty set is a subset of any set. Now we need to check the condition for all $w\in \emptyset$, $|w|\le3$. We can rewrite this condition as "If $w\in \emptyset$, then $|w|\le 3$." Since the "If $w\in \emptyset$" part is satisfied by no $w$, the premise is false and an implication with a false premise is always true. We might as well say "If $w\in\emptyset$, then $w$ is a sandwich," which is likewise true.