Question

1.2.14. Suppose S is a finite or countable set. Is it possible that P({s}) = 0 for every single s ∈ S? Why or why not?

Answer 1

Let S be a finite set.

Then by definition:

∃n∈N:|S|=n
where |S| denotes the cardinality of S.

From Cardinality of Power Set:

|P(S)|=2^n
where P(S) denotes the power set of S.

As n∈N it follows that 2^n ∈N and so P(S) is also by definition a finite set.
but P(S)=0
=>2^n = 0 (impossible) as n ∈ N
Thus P({s}) !=! 0