Characteristic function / Indicator function

Question

I really don’t understand the characteristic function in terms of sets. I get that it maps a set to the set {0,1} and you can use this to count the number of elements in that set. But,set operations like unions and intersections, I don’t get how it’s used, also proving the Inclusion-Exclusion principle. Could someone explain this to me, can’t find a site that explains it properly.

Answer 1

The characteristic (or : indicator) function for a subset $A$ of $X$ is the function :

$1_A ; X \to \{ 0,1 \} \text { such that : for every } x \in X : 1_A(x)=1 \text { iff } x \in A$.

But a function is a set of pairs, i.e. a subset of the cartesian product.

Thus :

$1_A = \{ (z,b) \mid z \in A \text { and } b \in \{ 0,1 \} \}$

and : $1_A \subseteq X \times \{ 0,1 \}$.

Answer 2

For definition of characteristic function see the answer of Mauro.

If $A,B$ are sets then $\mathbf1_{A\cap B}=\mathbf1_A\mathbf1_B$.

This because:$$x\in A\cap B\iff x\in A\wedge x\in B\iff\mathbf1_A(x)=1\wedge\mathbf1_B(x)=1\iff \mathbf1_A\mathbf1_B=1$$

Concerning unions we have: $$\mathbf1_{A\cup B}=\mathbf1_A+\mathbf1_B-\mathbf1_{A\cap B}=\mathbf1_A+\mathbf1_B-\mathbf1_{A}\mathbf1_{B}$$

Observe that by substitution of argument $x$ both sides we get $1$ as result if and only if $x\in A\cup B$, and $0$ otherwise. So the functions on LHS and RHS are the same.

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The principle of inclusion/exclusion rests on the equality:$$\mathbf1_{\bigcup_{i=1}^n A_i}=\sum_{i=1}^n\mathbf1_{A_i}-\sum_{1\leq i<j\leq n}\mathbf1_{A_i\cap A_j}+\cdots+(-1)^n\mathbf1_{A_1\cap\cdots\cap A_n}\tag1$$

For a proof of $(1)$ see this answer.

For any suitable measure $\mu$ we can take expectation on both sides of $(1)$ resulting in:$$\mu\left(\bigcup_{i=1}^n A_i\right)=\sum_{i=1}^n\mu(A_i)-\sum_{1\leq i<j\leq n}\mu(A_i\cap A_j)+\cdots+(-1)^n\mu(A_1\cap\cdots\cap A_n)$$