22
Apr

## Maths Proofs – lemmas and theorems

For all those serious Maths buffs … lemmas and theorems!

1. Lemma: All horses are of the same colour.

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 colour.

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 colour. 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 colour. Then the set of k+1 horses are all the same colour. We have k true => k+1 true; therefore all horses are of the same color.

2. 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 colour; and by the Lemma, it doesnt exist.