Which Sets Are Countably Infinite? (Multiple Answers)

(1) \(\{2, 4, 6, 8, \ldots\}\) — the even natural numbers

(2) \(\{1, 4, 9, 16, \ldots\}\) — the perfect squares

(3) \(\{1, 8, 27, 64, \ldots\}\) — the perfect cubes

(4) \(\mathbb{Z}\) — the set of all integers

Explanation

Even numbers: bijection \(f(n)=2n\). Squares: bijection \(f(n)=n^2\). Cubes: bijection \(f(n)=n^3\). Integers \(\mathbb{Z}\): bijection \(f(n) = 2n\) for \(n>0\) and \(f(n)=-2n+1\) for \(n \le 0\). All these are countably infinite sets with cardinality \(\aleph_0\). Points on a circle are uncountable (equivalent to an interval in \(\mathbb{R}\)), cardinality \(c\).