Question

show
\[\mathbb N\] contains an infinite number of pairwise disjoint infinite subsets

Answer 1

For prime numbers p, let A_p = { p^n | n in N }
e.g., A_2 = { 2, 2^2, 2^3, ....} = {2, 4, 8, 16, ...}
A_3 = { 3, 3^2, 3^3, ...} = (3, 9, 27, 81, ...}

Then it is clear that for any primes p and q not equal
\[A_p \cap A_q = \emptyset \]

Answer 2

...and of course each A_p has an infinite number of members.

Answer 3

thank you