Indicators and the inclusion-exclusion formula let ia be

Question

Indicators and the inclusion-exclusion formula. Let I_A be the indicator of A. Show the following:

a) The indicator of A^c, the complement of A, is I_Ac = 1 - I_A:

b) The indicator of the I intersection A_B of A and B is the product of I_A and IB IAB = I_AI_B:

c) For any collection of event A1,  .. . An, the indicator of their union is I_A1UA₂U...U_An = 1 -(1 -I_A1)(1 - I_A2)  . ..(I - I_An)

Expand the product in the last formula and use the rules of expectation to derive the inclusion-exclusion formula of Exercise 1.3.12.