I've got quite a tricky problem here:
Find an example of four infinite subsets of the set of all positive integers so that the intersection of any three of them is an infinite set, while the intersection of all four of them is an empty set.
I've got quite a tricky problem here:
Find an example of four infinite subsets of the set of all positive integers so that the intersection of any three of them is an infinite set, while the intersection of all four of them is an empty set.
Try these.
In the following sets definitions each of k, m, & n is a nonnegative integer and at least one is not zero.
$\displaystyle A = \left\{ {2^k \cdot 3^m \cdot 5^n } \right\}\quad \& \quad B = \left\{ {2^k \cdot 3^m \cdot 7^n } \right\}$
$\displaystyle C = \left\{ {2^k \cdot 5^m \cdot 7^n } \right\}\quad \& \quad D = \left\{ {3^k \cdot 5^m \cdot 7^n } \right\}$
Why do these work?