My younger brother(age: 14 years 7 months) and his classmates were given a set of eight questions by his class-teacher, which included the following two questions:
(i) Find, if you can, the fallacy in the following proof(!):
Claim:"any two positive integers are equal"
Proof: Suppose, $A_n$ be the statement
if $a$ and $b$ are any two positive integers,such that, $\max(a,b)=n$, then $a=b$
Suppose $A_r$ to be true. Let $a$ and $b$ be any two positive integers, such that $\max(a,b)=r+1$. Consider the two integers, $c=a-1$, $d=b-1$; then $\max(c,d)=r$. Hence $c=d$, for we are assuming $A_r$ to be true. It follows $a=b$; hence $A_{(r+1)}$ is true. $A_1$ is obviously true, for if $\max(a,b)=1$, since $a,b$ are by hypothesis positive integers, they must be both equal to $1$. Therefore, by mathematical induction , $A_n$ is true for every $n$. Now,if $a$ and $b$ are any two positive integers whatsoever, denote $\max(a,b)$ by $r$. Since $A_n$ has been shown to be true for every $n$, in particular, $A_r$ is true. Hence $a=b$.
What I have done:
If $a=1$,$b=1$, then $c=d=0$. So, $\max(c,d)$ is not defined, because $0$ is not a positive integer. So, $A_1$ is false, and hence, our basis step in the induction process suffers a heavy blow, which shatters the foundation upon which the palace is built, consequently bringing about utter destruction of the palace itself.
What's my problem:
Is there any simpler way to convince my brother how to proceed regarding disproving this claim?
Close. $A_1$ is true, not false. The error is in the inductive step. When you consider the two numbers $c=a−1,d=b−1$, there is no guarantee that these will be positive. For example, $1$ and $2$ have $\max\{1,2\}=2=1+1=r+1$ but $c=a−1=0$ is not a positive integer, so you cannot apply the induction hypothesis.