Use Zorn’s lemma to prove that, given sets A and B, if there is a surjection f : A → B then there is an injection g : B → A. (We did the dual problem, “injectiongives-surjection,” in class; this direction, though, requires the axiom of choice!) • HINT: Call a partial injection an injection h : C → A with C ⊆ B. For partial injections h1, h2, say that h1 / h2 if: 1. dom(h1) ( dom(h2), and 2. for x ∈ dom(h1), we have h1(x) = h2(x). Intuitively, h1 / h2 if h2 “extends” h1. Let Inj be the set of partial injections; show that (Inj, /) is a partial order satisfying the condition for Zorn’s lemma, and that a maximal element of (Inj, /) corresponds to genuine injection from B to A.
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