  We study the factorization and monotonicity method for inverse acoustic
scattering problems. Firstly, we give a new general functional analysis theorem
for the monotonicity method. Comparing with the factorization method, the
general theorem of the monotonicity generates reconstruction schemes under
weaker {\it a priori} assumptions for unknown targets, and can directly deal
with mixed problems so that the unknown targets have several different boundary
conditions. Using the general theorem, we give the reconstruction scheme for
the mixed crack that the Dirichlet boundary condition is imposed on one side of
the crack and the Neumann boundary condition on the other side, which is a new
extension of monotonicity method.
