1. (30 points) Find the error in the following proof that all horses are the same color. CLAIM: In any set of h horses, all horses are the same color. PROOF: By induction on h. Basis: For h = 1. In any set containing just one horse, all horses clearly are the same color. Induction step: For K 2 1, assume that the claim is true for h = k and prove that it is true for h=k+1. Take any set Hof k + 1 horses. We show that all the horses in this set are the same color. Remove one horse from this set to obtain the set Hy with just k horses. By the induction hypothesis, all the horses in Hy are the same color. Now replace the removed horse and remove a different one to obtain the set H2. By the same argument, all the horses in H2 are the same color. Therefore, all the horses in H must be the same color, and the proof is complete