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 � 1 assume that the claim is true for h = k and prove that it is true for h = k + 1 . Take any set H of k + 1 horses. We show that all the horses in the set are the same color. Remove one horse from this set to obtain the set H1 with just k horses. By the
induction hypothesis, all the horses in H1 are the same color. Now replace the removed horse and remove a di�erent 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.
"...prove that it is true for h = k + 1 . Take any set H of k + 1 horses. We show that all the horses in the set are the same color. Remove one horse from this set to obtain the set H1 with just k horses...."
The induction process requires to prove that given the proposition is true for k, then k+1 is true.
The above prove is proving that given k+1 is true, then k is true. So k decreases, and does not help to prove that k+2 ... is true.
The error in the proof is in the induction step. The proof assumes that if a set of k horses has the property that all the horses in it are the same color, then a set of k+1 horses will also have this property. However, this assumption is not correct.
To see why this is not true, consider a specific example. Let's say we have a set of two horses, one black and one white. According to the induction hypothesis, since there is only one horse in the set, it is true that all horses in the set are the same color (which is true in this case since there is only one horse).
Now, when we try to apply the induction step to a set of three horses, we remove one horse and obtain a set of two horses. According to the induction hypothesis, all the horses in this set (the remaining two horses) are the same color. But as we saw in the specific example, this is not true, because we started with a set containing a black and a white horse.
Therefore, the proof is incorrect, and the claim that all horses are the same color is not proven.