Discrete Mathematics

Question

I don’t really know where to even begin here. My first thought was that the intersection of j and k would be the universal set, but I don’t have any proof for that. I don’t have much practice with indexed families of sets, but I’ve done a lot with set theory this semester. I’m a third year math minor and this class is supposed to help transition you to more rigorous, proof-based math. Thanks for the help !

Answer 1

A_n is the set of all natural numbers greater than or equal to n. Part (a) asks whether the intersection of A_j and A_k must always be non-empty. A moment's reflection will show that this is true since the intersection of A_j and A_k will be A_j if j > k, or A_k otherwise, and no A_n is empty. Part (b) asks whether the intersection of all A_n is empty or not. The intersection of all A_n would be the set of all natural numbers greater than or equal to the "greatest natural number". Of course, there is no "greatest natural number", so there is nothing in the set; the intersection of all A_n is empty because every natural number k is missing from at least one of the sets (A_k+1).