False theorem: All horses are the same color.

Proof by induction:

P(n) is the statement: In every set of horses of size n, all n horses are the same color.
Base Case or P(1): One horse is the same color as itself. This is true by inspection.
Induction Step: Assume P(k) for some k≥1.
Proof of P(k+1):
Since {H1,H2,...,Hn} is a set of n horses, the induction hypothesis applies to this set. Thus, all the horses in this set are the same color.
Since {H2,H3,...,Hn+1} is also a set of n horses, the induction step likewise holds for this set. Thus, all the horses in this set are the same color too.

Therefore, all n+1 horses in {H1,H2,H3,...,Hn,Hn+1} are the same color. QED.



출처는: https://math.stackexchange.com/questions/428151/questions-on-all-horse-are-the-same-color-proof-by-complete-induction




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