Extra Credit Problem #4

Problem: Explain the logical flaw in the inductive proof given below that all horses are the same color.

To get full credit you must explain precisely where and why the proof goes wrong and you must express yourself well. What is logically wrong?

Due Date: Friday April 8^st.

Credit: Four points towards a quiz grade.

THEOREM: All horses are the same color.

PROOF: We will show that for any integer n, any set consisting of n horses contains horses of one color. (i.e. no matter which set of n horses we pick, all n of those horses are the same color -- which color it is can depend on which set we pick, of course.) We use mathematical induction on n.

Basis Step When n=1 we have the statement "any set consisting of just one horse" contains horses all of the same color, clearly true statement.

Induction Hypothesis. We make the inductive assumption that the statement is true when n=k, i.e. Any set comprised of k horses contains horses of only one color. We need to show that the statement is true when n=k+1, i.e. any set consisting of k+1 horses contains horses all of one color. Here's how we do it:

Consider an arbitrary set of k+1 horses. Say H1, H2, H3, ... , H(k+1).

The set {H1,H2,H3, ... , Hk} has cardinality k and so all these horses are of the same color, say color 1. Similarly, the set {H2,H3, ... , Hk, H(k+1)} has cardinality k and so all these horses are of the same color, say color 2. But color 1 and color 2 must in fact be the same color because all horses H2, H3, ... , Hk are in both sets. We conclude that all k+1 horses are the same color, as required. Therefore by the Principle of Mathematical Induction, all horses are the same color!!

QED