Explain what is wrong with the following proof by mathematical induction that al
ID: 2902831 • Letter: E
Question
Explain what is wrong with the following proof by mathematical induction that all horses are the same color: Clearly all horses in any set of 1 horse are all the same color: CLearly all horses in any set of 1 horse are all the same color. This completes the basis step. Now assume that all horses in any set of n horses are the same color. Consider a set of n+1 horses, labeled with the integers 1,2,...,n+1. By the induction hypothesis, horses 1,2,....,n are all the same color, as are horses 2,3,....,n,n+1. Because these two sets of horses have common members, namely, horses 2,3,4,...,n,all n+1 horses must be the same color. This completes the induction argument
Explanation / Answer
This proof is obviously wrong, because all horses are not of same color. So, where lies the fallacy?
Fallacy in the proof
The argument above makes the implicit assumption that the two subsets of horses to which the induction assumption is applied have a common element. This is not true when the original set (prior to either removal) only contains two horses.
Let the two horses be horse A and horse B. When horse A is removed, it is true that the remaining horses in the set are the same color (only horse B remains). If horse B is removed instead, this leaves a different set containing only horse A, which may or may not be the same color as horse B.
The problem in the argument is the assumption that because each of these two sets contains only one color of horses, the original set also contained only one color of horses. Because there are no common elements (horses) in the two sets, it is unknown whether the two horses share the same color. The proof forms a falsidical paradox; it seems to show something manifestly false by valid reasoning, but in fact the reasoning is flawed.
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