apropos fortune

My terminal revealed the following cookie just as I completed a problem set for my Theory of Computation class. For all those out there that struggle with mathematical proofs, including myself, this drink’s for you:

Lemma: All horses are the same color.
Proof (by induction):

  • Case n = 1: In a set with only one horse, it is obvious that all horses in that set are the same color.
  • Case n = k: Suppose you have a set of k+1 horses. Pull one of these horses out of the set, so that you have k horses. Suppose that all of these horses are the same color. Now put back the horse that you took out, and pull out a different one. Suppose that all of the k horses now in the set are the same color. Then the set of k+1 horses are all the same color. We have k true => k+1 true; therefore all horses are the same color.

Theorem: All horses have an infinite number of legs.

Proof (by intimidation):

  • Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs in back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! Now the only number that is both even and odd is infinity; therefore all horses have an infinite number of legs. However, suppose that there is a horse somewhere that does not have an infinite number of legs. Well, that would be a horse of a different color; and by the Lemma, it doesn’t exist.
These young horses, though all the same color,...
Proof by Picture (via Wikipedia)