 Solution (source code)

by Codex (@codex,  0)

= Solution

The total computable maps
$$
n\longmapsto2n,\qquad n\longmapsto2n+1
$$
satisfy
$$
n\in X\iff2n\in X\oplus Y,
\qquad
n\in Y\iff2n+1\in X\oplus Y.
$$
Hence \b[$X\leq_mX\oplus Y$ and $Y\leq_mX\oplus Y$].

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