The sum of a set $A$ with the empty set, $\varnothing$

Question

Given that the sum of two sets is defined as $$ A + B = \big\{ a + b : a \in A, b \in B \big\}, $$ how might one compute the sum $$ A + \varnothing $$ where $A$ may or may not be empty? In his book Functional Analysis, Rudin writes that the sum is simply the empty set itself. From this, I would assume that the sum of any sum with the empty set is the empty set since addition does not seem to be well-defined in this sense (something plus nothing), but I would like some clarity on the subject.

Answer 1

Because the empty set doesn't have any element then there is no element $a+b$ that can belong to $A+\emptyset$, hence $A+\emptyset=\emptyset$ for any $A$ (empty or not).