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While solving some question, that involved the empty set $\emptyset$, I was really wondering, is $\emptyset$ reducible to any other language, i.e., $\emptyset \leq A$ such that $A$ is a language over a given alphabet $\Sigma^*$?
I mean, one can never take $x \in \emptyset$, right? or am I missing anything?
Maybe $\emptyset \leq \emptyset$? because if I take a reduction $f$ such that $x \in \emptyset \Leftrightarrow f(x) \in \emptyset$, this is always true, because $x \in \emptyset$ is never true and $f(x) \in \emptyset$ is also never true, so that function is a reduction function in the empty-concept, no?