Chapter 12.5, Problem 38E

DISCUSS: All Cats Are Black? What is wrong with the following “proof” by mathematical induction that all cats arc black? Let P(n) denote the statement “In any group of n cats, if one cat is black, then they are all black.”

Step 1 The statement is clearly true for n = 1.

Step 2 Suppose that P(k) is true. We show that P(k + 1) is true.

Suppose we have a group of k + 1 cats, one of whom is black; call this cat “Tadpole.” Remove some other cat (call it “Sparky”) from the group. We are left with k cats, one of whom (Tadpole) is black, so by the induction hypothesis, all k of these are black. Now put Sparky back in the group and take out Tadpole. We again have a group of k cats, all of whom—except possibly Sparky—are black. Then by the induction hypothesis, Sparky must be black too. So all k + 1 cats in the original group are black. Thus by induction P(n) is true for all n. Since everyone has seen at least one black cat, it follows that all cats are black.