(i) If g is a function from Y to X such that gf is the identity on X, then f is one-to-one and g maps Y onto X.
(ii) A necessary and sufficient condition that f(A ∩ B) = f(A) ∩ f(B) for all subsets A and B of X is that f be one-to-one.
(iii) A necessary and sufficient condition that f(X - A) ⊂ Y - f(A) for all subsets A of X is that f be one-to-one.
(iv) A necessary and sufficient condition that Y - f(A) ⊂ f(X - A) for all subsets A of X is that f map X onto Y.
Exercise: Prove that if n is a natural number, then n ≠ n⁺, if n ≠ 0, then n = m⁺ for some natural number m. Prove that ω is transitive. Prove that E is a non-empty subset of some natural number, then there exists an element k in E such that k ∈ m whenever m is an element of E distinct from k.
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