3. What is wrong with the “proof” that all horses have the same color? Let P(n)

Question

3. What is wrong with the “proof” that all horses have the same color?

Let P(n) be the proposition that all the horses in a set of n horses are the same color. Basis step: Clearly, P(1) holds.

Inductive step: Assume that P(k) is true, so that all the horses in any set of k horses are the same color. Consider any k+1 horses; number these as horses 1, 2, . . . , k, k + 1. Now the first k of these horses all must have the same color, and the last k of these must also have the same color. Because the set fo the first k horses and the set of the last k horses overlap, all k+1 must be the same color. This shows that P(k+1) is true and finishes the proof by induction.

Explanation / Answer

here you can't say this by directly

the first k of these horses all must have the same color, and the last k of these must also have the same color

here we have to prove  k+1 must be the same color which is not possible by directli saying

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