Proof by induction with the Union of sets

Proof by induction:

$$P( \cup _{i=1}^n A_i)=\sum_{i=1}^n P(A_i) - \sum_{1 \leq i_1 < i_2 \leq n} P(A_{i_1} \cap A _{i2} ) + \sum_{1 \leq i_1 < i_2 <i_3 \leq n} P(A_{i1} \cap A_{i2} \cap A_{i3}) -...+(-1)^{n+1}P(A_1 \cap A_2 \cap ... \cap A_n)$$

I don't really understand the whole i1

• proofwiki.org/wiki/Inclusion-Exclusion_Principle – d.k.o. Jul 13 '15 at 9:56
• for two sets it is obvious $P\left( A\cup B \right) =P\left( A \right) +P\left( B \right) -P\left( A\cap B \right)$ – haqnatural Jul 13 '15 at 9:57
• What is $P$? My first association was power set, which doesn't fit here. – MvG Jul 13 '15 at 9:59
• @MvG I would assume $P$robability. – DRF Jul 13 '15 at 10:01