The birthday problem concept question

[maiad]

Let A(i) be the event that the first ith person have different birthdays for i=1,2,3...,n.
We note that A(i+1) is a subset of Ai such that Ai(A(i+1))= A(i+1)

I wonder why A(i+1) is a subset of Ai. If the first 3 people have no birthdays in common, shouldn't that also mean the first 2 people doesn't either? By that logic, shouldn't Ai be the subset of A(i+1)?

 

[mathman]

Your reasoning is correct up to the conclusion. Ai includes i+1 people who have same birthday as one of the first i, as well as all of A(i+1), therefore Ai is the bigger set.