Question

do anyone knows how to prove Inclusion and Exclusion Principle by mathematical induction ?

Answer 1

google does
http://planetmath.org/encyclopedia/PrincipleOfInclusionExclusionProof.html

Answer 2

looks largely the same here, so i think this is what you need
http://www.proofwiki.org/wiki/Inclusion-Exclusion_Principle

Answer 3

\[p(E1 U E2 U ...... UEn)= \sum_{i=1}^{n} - \sum_{i 1< i 2}^{} P (Ei1 n Ei2)+ . . . + (-1)^{r+1} \sum_{i1< i2 <.... < ir}^{} + ....+ (-1)^{n+1} P (E1 n E2n....n En)\]