Sum of an empty set and a finite set. Can we define the sum of an empty set and a finite set$?$
For example -
If $A= \{1,2\} , B= \emptyset$
Then what is $A+B$.
My intuition says it should be $A$.
But I couldn't find any proper reason behind it.

Some thing that you aren't considering is that you are using some "Algebra" inside your reasoning.
Remember that Set Theory is pure sets and their basic operations: Union, Intersection and Difference, all of them defined only by the relation $\mathbb \in$ and nothing else. And you mustn't confuse "set difference" with "number difference". Although numbers are also sets, both concepts are quite different. A casual study on formal set theory can make clearer this concepts.
In the other hand a "sum" is however an operation and operations over sets are not simply sets anymore, they become in an Algebraic Structure, then it is completely valid to ask what happens with B+ $\emptyset$ whenever A is a set, B and $\emptyset$ subsets of A, "+" an operation over A, and the answer is up there in the Abhijeet answer.
