# How to show that the complement of ATM $\leq_{m}$ L = {<M> : |L(M)| = 2}?

My original intention was to prove that $$L = \{\langle M \rangle \mid |L(M)| = 2 \}$$ is not turing recognizable but soon I realized that I could use the complement of ATM because the complement of ATM is not turing recognizable. And there's a corollary that if $$A \leq_{m} B$$ and $$A$$ is not turing recognizable then $$B$$ is not turing recognizable. In this case we could prove that $$L$$ is not turing recognizable.

However I have trouble proving that the complement of ATM $$\leq_{m} L = \{\langle M \rangle \mid |L(M)| = 2 \}$$, which stands for the complement of ATM is mapping reducible to $$L$$.

• Are the hints in this answer helpful to you? – John L. Mar 4 '19 at 4:50