Minimum Unsatisfiable Core
First thing to say that the minimality here is subset-minimality (as opposed to cardinality-minimality). Observe that minimality is not actually needed. The main objective is to block the assignment A, which led to the conflict. However minimality is beneficial because like this you block other "bad" assignments as well. 

Similarly, multiple MUCs are not a problem. All we need is to block the  assignment A so picking an arbitrary MUC is sufficient. Note also that the number of MUCs can be exponential, so going through all of them might not even be feasible and typically is not done in this context.

To compute an MUC there're various algorithms. The easiest is to go through each literal L in the set and remove it. If the set is still UNSAT, then move onto the next literal. If the set is SAT, put the literal L back (it's needed).  If I may shamelessly plug in my paper, here's more algorithms [1].

[1] Minimal Sets over Monotone Predicates in Boolean Formulae. J Marques-Silva, M Janota, A Belov. CAV13