Problem 75. Prove the conjecture in Part 13.b for an arbitrary positive integer m without appealing to the general product principle. (h)

(4) Bogart, #75
Prove the following statement using induction on m. Do NOT use the general product principle.

Theorem. There are nm functions from [m] = {1,2, m} to [n] = {1,2, n} for n and m positive integers.

Problem 77. Suppose that f is a function on the nonnegative integers such that f (0) = 0 and f (n) = n + f (n - 1). . Prove that f (n) = n (n + 1)/2. Notice that this gives a third proof that 1 + 2 + + n = n (n + 1)/2, because this sum satisfies the two conditions for f. (The sum has no terms and is thus 0 when n = 0.)

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