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.
 A: You couldn't find any proper reason behind it because there isn't a general definition of "set sum" that makes sense for all sets. The only reason why you think that:
$$\{1,2\} + \varnothing = \{1,2\}$$
is because you think that we are simply adding the elements of both sets together and since $\varnothing$ is empty, it simply gets added to each element of $\{1,2\}$ as $0$. However, I have a few questions for you to address if that's what you're going with;

*

*Why does it have to be the case that $0$ is added to each element of $\{1,2\}$?


*Why are we considering addition on $\mathbb{N}$ or whatever? What's so special? It might be very natural but how do you add the elements of one set to the "elements" of a set that is empty?
My point is that when we consider two sets $A,B \subseteq \mathbb{R}$, then there's a very natural way to define the set sum:
$$A+B = \{a+b \in \mathbb{R}: a \in A \land b \in B \}$$
But that's only because we have a clear notion of addition on $\mathbb{R}$. In general, this notion might not be available to us.
Taking this up a notch, how would you add, say, $\{\varnothing\}$ and $\varnothing$? Both of these are definitely sets, with one being a finite set and one being the empty set. What would be your definition of addition in this case?
Bottom line; you must suspend any intuition you might have about addition from school and, instead, define explicitly what you mean by "addition of sets". This is no different from carefully defining the set intersection and set union.
A: If by $A+B$ you mean $\{\,a+b\mid a\in A,b\in B\,\}$, then if $B$ is empty there is no $b\in B$ so there are no elements in $A+B$. "$B$ is empty" is very different from "zero is in $B$".
A: 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.
