Why is this relation irreflexive? And how can I prove it? Why is the relation R on A irreflexive if and only if ΔA ∩ R = ∅? I always thought the empty set is reflexive (and transitive, symmetric because it is vacuously true.)

By definition, $R$ is irreflexive iff for all $a\in A$ we have $(a,a)\
otin R$. This precisely says that the diagonal $\Delta A=\\{\,(a,a)\mid a\in A\,\\}$ is disjoint from $R$. Note that according to this, the empty relation is irreflexive, and it ias also (vacuously) symmetric and transitive. But the empty relation is reflexive only as relation on the empty set (i.e. $R=\emptyset$ is reflexive iff $A=\emptyset$).