Suppose , I have two objects A and B. A has a set of properties P and B has a set of properties Q. It is possible that when I combine A and B the combination has a property c that is neither in P nor in Q (i.e c ∉ P∪Q ). Examples are everywhere so i am not feeling the need to provide any. Now my question is : what exactly is property and where do new properties come from?
Your profile mentions an interest in math, which you seem to be pursuing in the formulation of your question. So let me suggest how physics approaches this same kind of question. Terminology-wise, your "object" is a "system", and its "properties" are the system's "observables".
Then the collection of all possible states a system can assume is modelled by (the unit vectors of) a Hilbert space H corresponding to that system. And then the collection of bounded Hermitian operators B(H) on H represent all the system's observables, and this B(H) comprises what's called a C*-Algebra.
Now, if you have two systems, A and B represented by H_A and H_B, then the composite system is represented by H_A x H_B, and the C*-Algebra of observables B(H_A x H_B) of that composite system is way more complicated than B(H_A) and B(H_B) alone (and way more complicated than their simple set-theoretic union).
Now, regarding the comments, a property like "wetness" wouldn't really be a formal observable in the above sense (at least I'm pretty sure -- make that almost absolutely positive -- not). But if you're looking for a rigorous mathematical treatment of your question, it couldn't hurt (except for the mental time and effort) to google some of the above stuff and start with that kind of approach, which has been carefully developed by many people during the last ~75 years (I'm dating it since Irving Segal's 1947 seminal paper).