I'm wondering if this holds $$\overline{(A\cup \overline{B})}=\overline{A}\cap B$$
I have this problem in probability theory, if $A$ and $\overline{B}$ are independent events then $\overline{A} $ and $B$ are also independent events. Can I use De Morgan's law to approach this?
Since $A, B$ are independent you have that $$P(A)P(B^c)=P(A)\left(1-P(B)\right)=P(A)-P(A)P(B)=P(A)-P(A\cap B)=P(A\cap B^c)$$ since $$A-(A\cap B)=A\cap B^c$$ so actually De Morgan's law is not very helpful in this context. It was helpful to note that $$A=(A\cap B) \cup (A \cap B^c )$$ so that $$A-(A\cap B)=A\cap B^c$$
