-1

The following statement is false, but still, a proof is given. I need to determine the error.

"In any set of $n≥1$ circles in $R^2$, all circles have the same radius".

Base case: If $n=1$, then every circle in the set has the same radius, so the statement holds.

Induction hypothesis: Suppose that the statement is true for $n$, i.e., that any $n$ circles in $R^2$ have the same radius.

Induction step. Let us show that any $n+1$ circles in $R^2$ have the same radius. Let $C_1$, $C_2$, $...$, $C_n$, $C_{n+1}$ be any circles in $R^2$. By the induction hypothesis, we have that $C_1$, $C_2$, $...$, $C_n$ have the same radius. Also by the induction hypothesis, we have that $C_2$, $C_3$, $...$, $C_n$, $C_{n+1}$ have the same radius. Therefore, all circles $C_1$, $C_2$, $...$, $C_n$, $C_{n+1}$ have the same radius. Hence, any $n$ circles in $R^2$ have the same radius.

To get started, I'm not even sure I understand the statement clearly. It says that, in any set of $n≥1$ circles in $R^2$, all of them have the same radius. But the induction hypothesis is claiming that any $n$ circles in $R^2$ have the same radius, which is not the same, because as I understand, the statement is only saying that, if we take an arbitrary set of $n$ circles, then we'll find that all of them have the same radius. If that's so, the error must be in the induction hypothesis, but I'm not sure. Can you please help me?

lisa
  • 1
  • 4
    Does this answer your question? Show that all horses are of the same color. - I recalled this basically same "issue" regarding proving that all horses have the same color, so I did a site search to find there being $62$ questions found, with the proposed duplicate just being the first one on the list. I also just noticed that Questions on "All Horse are the Same Color" Proof by Complete Induction is in the ... – John Omielan Nov 20 '23 at 01:17
  • (cont.) "Related" section on the right side, with this other question possibly being a better duplicate one. Last, but not least, welcome to Math SE. – John Omielan Nov 20 '23 at 01:22
  • 1
    You have only established that one circle $C_1$ has the same radius. When you suppose it is true for $k$ circles, you are assuming that $k\gt 1$, i.e. that circle $C_2$ with the same radius as circle $C_1$ exists. This is why you must prove that the radius of $C_n$ equals the radius of $C_1$ implies that the radius of circle $C_{n+1}$ has the same radius. From that you would conclude that $C_2$ has the same radius, $C_3$ has the same radius, etc. You haven't done that and can't because the theorem is false. – John Douma Nov 20 '23 at 01:37

1 Answers1

0

In the induction step, when you say "$C_{2},C_{3},\cdots,C_{n},C_{n+1}$", you are assuming that $n \geq 2$, not only $n\geq1$, you can see the proof fails if you consider $n=2$.

Raul Fernandes Horta
  • 604