Formula for the number of elements in exactly one of $n$ sets is $N_1 = S_1 - 2S_2 + 3S_3 - \ldots \pm nS_n$ where $S_i$ is the sum of cardinalities of $i$-wise intersections except $S_1$ which is the sum of cardinalities of the given sets.
Let's consider an example for $n = 2$ with $\{1, 2, 3\}$ and $\{1, 2, 4\}$. By the formula above, we have $N_1 = 3 + 3 - (2\times 2) = 2.$ What does $N_1 = 2$ mean given that both $\{1, 2, 3\}$ and $\{1, 2, 4\}$ have cardinality three? If this is wrong (in any sense), which part of the formula am I looking at wrong? Thanks.
"The elements that are in exactly one set" doesn't mean "the elements of one specific set"; it means "the elements such that the number of sets they are in is equal to $1$". So in your example, those elements are $3$ and $4$: each of those is in one of the sets but not in the other. (On the other hand, $1$ and $2$ are not included, since they are each in both of the sets, not just one of them.) There are two such elements, so $N_1=2$.