[Math] Proof about Inclusion-exclusion formula

Question

The problem requires to use the indicate function to prove the Inclusion-exclusion formula. But I really don't know what to do.

Anyone can help with that? Thanks!

Answer 1

The indicator function $\mathbf 1_A$ is a random variable that can take on values $0$ and $1$ only. As the name implies, its value is $1$ exactly when the event $A$ occurs. Since $\mathbf 1_A$ takes on value $1$ with probability $P(A)$, its _expected value is$$E[\mathbf 1_A] = 0\times (1-P(A)) + 1\times P(A) = P(A).$$

So, begin by showing that $1 - \mathbf 1_A$ is the indicator function of $A^c$, and then argue that the event $A^c$ occurs if and only if the outcome is not a member of any of the $A_i$, that is, if and only if each and every one of the events $A_i^c$ occurs. Since $\mathbf 1_{C \cap D} = \mathbf 1_C\mathbf 1_D$ (prove this!), show that $$\mathbf 1_{A^c} = \prod_{i=1}^n \mathbf 1_{A_i^c} = \prod_{i=1}^n (1 - \mathbf 1_{A_i})$$