While we are thinking about sets, consider what happens to the additive principle when the sets are NOT disjoint. Suppose we want to find \(\card{A \cup B}\) and know that \(\card{A} = 10\) and \(\card{B} = 8\text{.}\) This is not enough information though. We do not know how many of the 8 elements in \(B\) are also elements of \(A\text{.}\) However, if we also know that \(\card{A \cap B} = 6\text{,}\) then we can say exactly how many elements are in \(A\text{,}\) and, of those, how many are in \(B\) and how many are not (6 of the 10 elements are in \(B\text{,}\) so 4 are in \(A\) but not in \(B\)). We could fill in a Venn diagram as follows:
in-context