Should the CV and statement of purpose for grad school be mutually exclusive, in particular if something is already in my CV, is it not necessary to mention it in the statement of purpose? For example, for a paper that I published, and I already included it in my CV, then I wouldn't need to say I published this paper in so-and-so journal in my statement of purpose?
Your CV is a straightforward and concise list of your academic credentials. Your statement of purpose is a chance for you to expand on your credentials (specifically those that make you an especially strong candidate, or to explain things that might seem to weaken your application) and also give some insight into your future plans.
They are not mutually exclusive, nor should they overlap entirely; they serve completely different purposes.
in particular if something is already in my CV, is it not necessary to mention it in the statement of purpose?
If you have something more to say about it that strengthens your application, go ahead and say it in your statement of purpose.