Existence, Uniqueness and Counterexamples · Medium problem

There is an integer $n$ such that every prime $p$ satisfies $p \le n$ and $p \ge 2n$.

There is an integer $n$ such that every prime $p$ satisfies $p \le n$ or $p \ge 2n$.

For every integer $n$ there is a prime $p$ with $p \le n$ or $p \ge 2n$.

There is an integer $n$ and a prime $p$ with $p \le n$ or $p \ge 2n$.