# If two languages are decidable, can one be mapping reducible to the other?

If I have two decidable languages $$A$$ and $$B$$, is $$A \leq_m B$$ true? How would I show this?

• What if $B = \emptyset$? – Yuval Filmus Oct 20 '20 at 23:03

That is false. Consider for example the alphabet $$\Sigma = \{0,1\}$$ and the two languages $$A = \emptyset$$ and $$B = \Sigma^*$$.
Clearly both $$A$$ and $$B$$ are decidable (by the trivial algorithms that always reject and always acccept, respectively) but it is false that $$A \le_m B$$.