Classify These Sets by Cardinality Type

(1) {1, 2, 3, ..., 50},Finite

(2) {2, 4, 6, 8, ...},Countably Infinite (ℵ₀)

(3) ℤ (all integers),Countably Infinite (ℵ₀)

(4) All points on the real line,Uncountable (≥ c)

(5) All subsets of ℕ,Uncountable (≥ c)

Explanation

\(\{1,...,50\}\) is finite (50 elements). Even numbers and \(\mathbb{Z}\) are countably infinite (\(\aleph_0\)). All points on the real line has cardinality \(c\). All subsets of \(\mathbb{N}\) (the power set \(P(\mathbb{N})\)) has cardinality \(2^{\aleph_0} = c\) — uncountable, since \(c > \aleph_0\).