# elementary set theory – Proving a set of statements is countable or uncountable?

Let the set $$S = { A_{1}, A_{2}, A_{3},… }$$ be a countably infinite set of statements. Let the set $$V$$ be defined as follows:

A) $$S subset V$$

b) For any $$alpha in V,$$ the negation of $$alpha$$ is also in V.

c) For any $$alpha, beta in V$$, $$(alpha vee beta) in V$$

d) $$V$$ does not contain anything else.

Determine whether $$V$$ is countable or uncountable (with proof).

Unfortunately I’m feeling kind of lost with this question and don’t really know where to start. My first thought was it may be countable since I was thinking of it as a union of countable sets, but I’m not sure if it is a countable union or not. Any guidance or a hint in the right direction would be much appreciated?