  We study a strongly attractive system of a few spin-1/2 fermions confined in
a one-dimensional harmonic trap, interacting via two-body contact potential.
Performing exact diagonalization of the Hamiltonian we analyze the ground state
and the thermal state of the system in terms of one-- and two--particle reduced
density matrices. We show how for strong attraction the correlated pairs emerge
in the system. We find that the fraction of correlated pairs depends on
temperature and we show that this dependence has universal properties analogous
to the gap function known from the theory of superconductivity. In contrast to
the standard approach based on the variational ansatz and/or perturbation
theory, our predictions are exact and are valid also in a strong attraction
limit. Our findings contribute to the understanding of strongly correlated
few-body systems and can be verified in current experiments on ultra-cold
atoms.
