1. To define the factorial function f (n) = n! by simple recursion, how can c and h in Corollary 1.3.3 be chosen? For h to serve this purpose, for which n and x (if any) are the values of h(n, x ) uniquely determined?

2. Let (X, ) be a well-ordered set. Let > be the reversed ordering as usual, that is, x > y means y <>x . Suppose that (X, >) is also well-ordered. Let f be a function from N onto X . Show that f cannot be one-to-one. Hint: If it is, then define h by recursion on N such that h(0) is the least element of X for <,>h(1) the next-least, and so on. The range of h has no largest element.

3. Given a partially ordered set (X, ≤), a subset A ⊂ X , and an element x ∈ A, x is called a minimal element of A iff there is no other y ∈ A with y <>x . Then (X, ≤) will be called min-ordered iff every non-empty subset of it has a minimal element. (Note that any well-ordered set is min-ordered.)

(a) Show that the induction principle (1.3.1) holds with "min-ordered" in place of "well-ordered."

(b) Do likewise for the recursion principle (1.3.2).