*** Entailment ***

(a) If Gamma |= phi and Delta |= phi then Gamma INT Delta |= phi

False.
Take, e.g.,
Gamma = {a}
Delta = {b}
phi = a | b
Clearly, Gamma |= phi and Delta |= phi, however Gamma INT Delta = {}.
Therefore it is not true that Gamma INT Delta |= phi, since phi is not a tautology.


(b) If Gamma |= phi and Delta |= phi then Gamma UNION Delta |= phi

True.
Models(Gamma UNION Delta) = Models(Gamma) INT Models(Delta) \subseteq Models(Gamma)
Besides, 
Models(Gamma) \subseteq Models(phi) because Gamma |= phi.
Therefore Models(Gamma UNION Delta) \subseteq Models(phi) q.e.d.


(c) If Gamma |= phi and Delta |# phi then Gamma UNION Delta |= phi

True. Same proof as in (b).


(d) If Gamma |# psi then Gamma |= ~psi.

False.
Take, e.g.,
Gamma = {a}
psi = b
Clearly, {a} |# b. However, {a} |# ~b.

(e) If Gamma |= ~psi then Gamma |# psi.

True.
Models(Gamma) \subseteq Models(~psi).
Models(psi) INT Models(~psi) = {}
Therefore Models(Gamma) INT Models(psi) = {}, then it is not true that Models(Gamma) \subseteq Models(psi) (unless Gamma is a contradiction, because in that case Models(Gamma)={} and Gamma entails both psi and ~psi).