Is the empty set always part of the result of an intersection? From very basic set theory we have that:
"The empty set is inevitably an element of every set."
Then, is it correct to assume that the intersection of 
$A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$ is actually $\{\emptyset, 3\}$, and not just $\{3\}$? 
Thank you

While it is true that for all sets $A$, we have $\emptyset \subseteq A$, it is not true that for all sets $A$, $\emptyset \in A$.
The intersection of $A,B$ is defined by $A \cap B = \{x | x \in A$ and $x \in B\}$.
What is important here is the difference between being a subset and being an element of a set. Do you now see why the empty set is not contained in $A\cap B$, with reference to your example?
