Injections in both directions yield a bijection — a result that does not require the axiom of choice. Proof by the Cantor-Bernstein alternating partition. Concrete use: |ℕ| = |ℕ×ℕ| via Cantor's pairing π(m,n) = (m+n)(m+n+1)/2 + n.
Injections in both directions yield a bijection — a result that does not require the axiom of choice. Proof by the Cantor-Bernstein alternating partition. Concrete use: |ℕ| = |ℕ×ℕ| via Cantor's pairing π(m,n) = (m+n)(m+n+1)/2 + n.