  One-dimensional spinless Bose and Fermi gases with contact interactions have
the close interrelation via Girardeau's Bose-Fermi mapping, leading to the
correspondences in their energy spectra and thermodynamics. However,
correlation functions are in general not identical between these systems. We
derive in both systems the exact universal relations for correlation functions,
which hold for any energy eigenstate and any statistical ensemble of the
eigenstates with or without a trapping potential. These relations include the
large-momentum behaviors of static structure factors and of momentum
distributions as well as energy relations, which connect the sums of kinetic
and interaction energies to the momentum distributions. The relations involve
two- and three-body contacts, which are the integrals of local pair and triad
correlations, respectively. We clarify how the relations for bosons and
fermions differ and are connected with each other. In particular, we find that
the three-body contact makes no contribution to the bosonic energy relation,
but it plays a crucial role in the fermionic one. In addition, we compute the
exact momentum distribution for any number of fermions in the unitary limit.
