I have understood that data science is part of AI. However, do we have some area in data science which we can say does not fall under AI? If yes, please name them.
I would not concern yourself too much about any structuring of knowledge that declares that one subject is categorised as one thing or another. These structures are often wrong and knowledge in general is more fluid and difficult to define than can be viewed as some kind of Venn diagram.
In addition, both Data Science and AI are poorly defined, and have more in common with marketing terms used to recruit staff and sell products than stricter academic subject areas such as Maths or Modern Languages. You will find Data Scientists working on "AI" projects and "AI Engineers" using basic statistics to analyse data.
Having said that, I would prefer to say that Data Science and AI share a common set of tools, especially when it comes to machine learning. Neither contains the other in my opinion.
I agree with what has been said by others.
To me, the two fields don't deal with the same object: A.I. is somehow the field of making intelligent machines & tools while data science's core objective consists in retrieving information and taking the most of available data.
You could therefore do data science all by yourself without any other tool than your brain, a sheet of paper and a pen, and this would not involve A.I. at all (but would certainly be painful).
Besides, creating intelligent machines & tools imply at some point those machines & tools to be able to deal with data and that's where the two fields find common ground.