Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify as...

Assign a grade of A (correct), C (partially correct), or F (failure) to each. Justify assignments of grades other than A.

_ (a) Claim. All horses have the same color.

“Proof.” We must show that for all in every set of n horses, all horses in the set have the same color. Clearly in every set containing exactly 1 horse, all horses have the same color.

Now suppose all horses in every set of n horses have the same color. Consider a set of n + 1 horses. If we remove one horse, the horses in the remaining set of n horses all have the same color. Now consider a set of n horses obtained by removing some other horse. All horses in this set have the same color. Therefore all horses in the set of n + 1 horses have the same color. By the PMI, the statement is true for every