91Ó°ÊÓ

Given that Proposition$4.4$.

We will apply mathematical induction to the number of sets. For$n=2$, see Proposition $4.3.$Now assume for $n=k$we have. $P\left(E_1∪E_2∪…∪E_k\right)=\underset{i=1}{\overset{k}{∑}}P\left(E_i\right)-\underset{i_1<i_2}{∑}P\left(E_{i_1}{E}_{i_2}\right)+…+(-1{)}^{r+1}\underset{i_1<…<i_r}{∑}P\left(E_{i_1}…E_{i_r}\right)+…+(-1{)}^{k+1}P\left(E_1…E_k\right)$

For $n=k+1.$Set $E=E_1∪…∪E_k$and apply Proposition $4.3.$Then we get

$P\left(E_1∪E_2∪…∪E_k∪E_{k+1}\right)=P\left(E\right)+P\left(E_{k+1}\right)-P\left(E{E}_{k+1}\right)$

Using the induction hypothesis we get

$P\left(E_1∪E_2∪…∪E_{k+1}\right)=P\left(E_{k+1}\right)+\underset{i=1}{\overset{k}{∑}}P\left(E_i\right)-\underset{i_1<i_2}{∑}P\left(E_{i_1}{E}_{i_2}\right)+…+(-1{)}^{r+1}\underset{i_1<…<i_r}{∑}P\left(E_{i_1}…E_{i_r}\right)+…+(-1{)}^{k+1}P\left(E_1…E_k\right)-P\left(E{E}_{k+1}\right)$

Now observe that

$$\begin{gathered} P\left(E{E}_{k+1}\right)=P\left(\left(E_1∪…∪E_k\right)E_{k+1}\right)=P\left(E_1{E}_{k+1}∪…∪E_k{E}_{k+1}\right) \\ =\underset{i=1}{\overset{k}{∑}}P\left(E_i{E}_{k+1}\right)-\underset{i_1<i_2}{∑}P\left(E_{i_1}{E}_{i_2}{E}_{k+1}\right)+…+(-1{)}^{r+1}\underset{i_1<…<i_r}{∑}P\left(E_{i_1}…E_{i_r}{E}_{k+1}\right)+…+(-1{)}^{k+1}P\left(E_1…E_k{E}_{k+1}\right) \end{gathered}$$

Combining this with the identity above we get

$P\left(E_1∪E_2∪…∪E_{k+1}\right)=\underset{i=1}{\overset{k+1}{∑}}P\left(E_i\right)-\underset{i_1<i_2}{∑}P\left(E_{i_1}{E}_{i_2}\right)+…+(-1{)}^{r+1}\underset{i_1<…<i_r}{∑}P\left(E_{i_1}…E_{i_r}\right)+…+(-1{)}^{k+2}P\left(E_1…E_{k+1}\right)$

Hence we get the desired identity by mathematical induction.