1.4

Exercise 1.4. Here is a fallacious proof using mathematical induction. Where does the proof go wrong? Everyone is the same age. We show this by showing that every group of people has the same age. In the base case, we consider a group of one person; clearly, that person is the same age as him- or herself. For the induction step, we assume that in any group of size n, ev- eryone is the same age. Now consider a group G of size n + 1. Let a and b be distinct people in G. The group G -a - that is, the group of people in G except for a - is a smaller group. And b is in G. Let r be someone in G; then clearly b and r are the same age. Now consider the group G -b _ the group of people in G except for b. Now r and a are both in G, So r and a are the same age. But since r and b are also the same age, a and b must be the same age. And since a and b were arbitrary, this holds for everyone in G. So everyone in QG is the same age. QED