Maybe worth noting that the backward implication also holds! So if the complements are independent, then so are the events themselves. If we let A = Xᶜ and B = Yᶜ be independent, then (by this vid) Aᶜ = Xᶜᶜ=X and Bᶜ=Yᶜᶜ = Y are also indpendent
Yes. If Ac and B are independent, so are A and B, A and Bc, and Ac and Bc. Independence of any one of those pairs implies independence of the others. Loosely, independence means that knowing that one event happened (or didn't happen) doesn't change the probability of the other event.
I confess to not being well versed in the full historical context (and I'm sure the knowledge that "not (A or B)" and "not A and not B" are the same thing goes *way* back, long before De Morgan), but in probability that notion is typically referred to as one of De Morgan's Laws.
