Quick Question about inclusion-Exclusion principle

Question

|S| indicates the cardinality of a set S (which may be considered as the number of elements of the set, if the set is finite).

I just learnt about this principle when trying to solve a problem and found myself messily applying this principle.

I don't understand why the cardinality is included in the principle. Does it not hold if you take all the | | out of the statement and replace with ( ), and then is that not a stronger principle and true?

Answer 1

It can be shown that: $$1_{\bigcup_{i=1}^n A_i}=\sum_{i=1}^n1_{A_i}-\sum_{1\leq i<j\leq n}1_{A_i\cap A_j}+\cdots+(-1)^{n-1}1_{A_1\cap\cdots\cap A_n}\tag1$$

where $1_B$ denotes the indicator function of set $B$.

Taking integrals wrt some measure $\mu$ on both sides we arrive at:$$\mu(\bigcup_{i=1}^n A_i)=\sum_{i=1}^n\mu(A_i)-\sum_{1\leq i<j\leq n}\mu(A_i\cap A_j)+\cdots+(-1)^{n-1}\mu(A_1\cap\cdots\cap A_n)$$

The inclusion/exclusion rule concerning cardinality shows up if $\mu$ is prescribed by $A\mapsto|A|$.

Care is needed because on the RHS we also meet subtraction. Under the condition that every $A_i$ is finite nothing can go wrong in the cardinality case.

Answer 2

You might be able to define addition and subtraction on "multisets", so if $A=\{2,3,5\}$ and $B=\{5,7,8\}$ then $A+B=\{2,3,5,5,7,8\}$ and $A+B-(A\cap B) = \{2,3,5,7,8\}$. Defining things in this way might have both advantages and disadvantages.