Let S be a subset of the reals such that S&cap;[a,b] and S<sup>c</sup>&cap;[a,b] cannot be written as a countable union of closed sets for any a&lt;b. This can be done (this <a href="http://planetmath.org/?op=getobj&from=objects&id=11351">explicit example of a non-Borel set</a> achieves this). Let &#x0211a; be the rationals. Then, A=(Sx&#x0211a;)U(S<sup>c</sup>x&#x0211a;<sup>c</sup>) and B=(Sx&#x0211a;<sup>c</sup>)U(S<sup>c</sup>x&#x0211a;) should do it.