# Inclusion and Exclusion proofs in RSA accumulators

I am reading about RSA accumulators from this video and few other sources. I had a confusion whether the complete set is needed to construct a proof of inclusion/exclusion in an RSA accumulator or is it possible to create the proof of inclusion/exclusion just by knowing the accumulator root and the value for which membership/non-membership proof has to be provided. Can someone please clarify this?

eg. If you have elements $$S_1 = \{x_1, x_2, ... x_k\}$$, the accumulator is $$A_{S_1} = g^{\prod_{i=0}^k x_i}$$. Now if $$y_1$$ is to be added, the set will become $$S_2 = \{x_1, x_2, ... x_k, y_1\}$$ and the accumulator will become $$A_{S_2} = g^{{\prod_{i=0}^k x_i} . y_1}$$. The witness for $$y_1$$ is $$A_{S_1}$$.