Find the mistake in the following proof that all dogs are the same color (if there's any):

Let's use induction. Consider groups of 1 dog. All dogs of every group are the same color, of course. So we now that it's true for 1. Suppose it's true for groups of k dogs, i.e. every group of k dogs are the same color. Then let's consider any group A of k+1 dogs. Consider a subgroup of A containing k dogs. Let's call x the dog in A but not in the subgroup. Then by induction, all dogs in the subgroup are the same color. Now consider a subroup of A of k dogs, with x in the subgroup. All dogs except for x are the same color. Then, since every group of k dogs are the same color (by induction), all dogs in A are the same color. So x and every dog are the same color.

Isn't it simpler to say that just because 1 particular group of k dogs are all the same colour doesn't mean all groups of k dogs are the same colour. Assuming it's true is assuming that the statement you are trying to prove is true. You can't prove something by assuming it's true. It's false logic.

Posted by Stuart on 2003-08-06 21:28:54