Explain what is wrong with the following proof by induction that all cows are br

Question

Explain what is wrong with the following proof by induction that all cows are brown. Let P(n) be the statement "any group of n cows have the same color." P(1) holds because a single cow has only one color. Now, assume P(n) holds for n greater than or equal to 1. We want to show P(n+1) holds. Let S = {c_1, c_2,...,c_(n+1)} be a group of n+1 cows. Since we are assuming P(n) holds, the group {c_1, c_2,...,c_n} must all be the same color and similarly {c_2,c_3,...,c_(n+1)} must all be the same color. Since the sets overlap, e.g. c_2 is in both sets, all n+1 cows in S must be the same color. So P(n+1) holds. Now, since we know brown cows exist (I've seen one with my own eyes), if we take the group of all cows in the world, they must all be brown.

Explanation / Answer

The proof is wrong because the base value of n for this case should be n=2....P(2) and not n=1.......P(1). The first step of proof that proofing that for n=1 ,it is true is a mistake and hence complete proof is wrong....

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