No need to watch any lecture.
Read Kenneth Rosen article on nested quantifiers and then do a related exercise on this on how to expand $R(x,y)$
Consider domain of x as {1,2} and Y as {a,b}
I can write $\forall x \exists y R(x,y)$
as
$(R(1,a) \lor R(1,b)) \land (R(2,a) \lor R(2,b))$
means I loop through all values of x, and in this loop I have another loop of y, and for each and every x, I must find atleast one y such that R(x,y) is true and I am done!
