Answer:
Explanation:
To construct an infinite set of sets of natural numbers such that all of their pairwise symmetric differences are infinite, we can define the following sequence of sets:
S_1 = {1, 2, 3, ...}
S_2 = {2, 3, 4, ...}
S_3 = {1, 3, 5, ...}
S_4 = {2, 4, 6, ...}
S_5 = {1, 4, 7, ...}
S_6 = {2, 5, 8, ...}
...
In general, S_n is the set of natural numbers that are congruent to n mod 2. For example, S_1 is the set of all odd natural numbers, S_2 is the set of all even natural numbers, and so on.
To show that all of their pairwise symmetric differences are infinite, we can consider any two distinct sets S_n and S_m. Without loss of generality, assume that n < m. Then, the symmetric difference between S_n and S_m is the set of natural numbers that are congruent to n or m mod 2, but not both:
S_n Δ S_m = {k ∈ N : (k ≡ n mod 2) XOR (k ≡ m mod 2)}
Since n < m, we can see that every other natural number is in this set, and therefore it is infinite. Hence, the sequence {S_n} is a set of sets of natural numbers such that all of their pairwise symmetric differences are infinite