Answer:
Produce an infinite collection of sets A1,A2,A3, . . . with the property that every Ai has an infinite number of elements, Ai ∩ Aj = ∅ for all i = j, and [infinity] i=1 Ai = N.
Explanation:
Solution
For n ∈ N,
define A_n = {2
^n−1 ,(3)(2n−1
),(5)(2^n−1
),(7)(2^n−1
), . . .}
I.e. A_n is all odd multiples of 2^n−1
. We must show that these sets satisfy the desired properties.
• (Infinite Number of Elements).
It is clear that the set A_n = {2
^n−1
,(3)(2^n−1
)(5)(2^n−1
),(7)(2^n−1
), . . .} has infinitely many elements.
• (Disjoint).
Given A_n and A_m with n ≠ m, we can assume, without loss of generality, that n < m. Suppose that there existed some x ∈ A_n ∩ A_m. Then by definition of these sets, there exists some odd numbers k and l such that x = 2^n−1 . k = 2^m−1 . l.
However since n < m, we have that n ≤ m − 1, and therefore we can write 2^m−1 = (2^n
)(2 i
) with i ≥ 0. Hence we have 2^n−1 . k = 2^n. 2
^i. l
Dividing both sides by 2^n−1 yields k = (2)(2^i
) .l, which contradicts the assumption that k is odd. Therefore A_n ∩ A_m = ∅.
• (Union is N).
We want to show that [infinity] i=1 A_n = N.
(⊆). Since each A_n is a subset of N, the union of these sets is a subset of N as well.
(⊇).Given any x ∈ N, we can write x = 2^n−1 . k for some n ∈ N where k is odd. Then x ∈ A_n, as desired.
