  We extend the formalism of weak interaction processes, obtaining new
expressions for the transition rates, which greatly facilitate numerical
calculations, both for neutrino-nucleus reactions and muon capture. We have
done a thorough study of exclusive (ground state) properties of $^{12}$B and
$^{12}$N within the projected quasiparticle random phase approximation (PQRPA).
Good agreement with experimental data is achieved in this way. The inclusive
neutrino/antineutrino ($\nu/\tilde{\nu}$) reactions $^{12}$C($\nu,e^-)^{12}$N
and $^{12}$C($\tilde{\nu},e^+)^{12}$B are calculated within both the PQRPA, and
the relativistic QRPA (RQRPA). It is found that the magnitudes of the resulting
cross-sections: i) are close to the sum-rule limit at low energy, but
significantly smaller than this limit at high energies both for $\nu$ and
$\tilde{\nu}$, ii) they steadily increase when the size of the configuration
space is augmented, and particulary for $\nu/\tilde{\nu}$ energies $> 200$ MeV,
and iii) converge for sufficiently large configuration space and final state
spin.
