Question

What is wrong with this “proof”? “Theorem” For every positive integer n, if x and y are positive integers with $\mathbf{max}\mathbf{}\mathbf{(}\mathit{x}\mathit{,}\mathit{y}\mathbf{)}\mathbf{=}\mathit{n}$, then$\mathit{x}\mathbf{=}\mathit{y}$

Basis Step: Suppose that $\mathit{n}\mathbf{=}\mathbf{1}$. If $\mathbf{max}\mathbf{}\mathbf{(}\mathit{x}\mathit{,}\mathit{y}\mathbf{)}\mathbf{=}\mathbf{1}\mathbf{}$and x and y are positive integers, we have $\mathit{x}\mathbf{=}\mathbf{1}$and $\mathit{y}\mathbf{=}\mathbf{1}$.

Inductive Step: Let k be a positive integer. Assume that whenever $\mathbf{max}\mathbf{}\mathbf{(}\mathit{x}\mathit{,}\mathit{y}\mathbf{)}\mathbf{=}\mathit{k}\mathbf{}$and x and y are positive integers, then x = y. Now let $\mathbf{max}\mathbf{}\mathbf{(}\mathit{x}\mathit{,}\mathit{y}\mathbf{)}\mathbf{=}\mathit{k}\mathbf{+}\mathbf{1}\mathbf{}$, where x and y are positive integers. Then , so by the inductive hypothesis, $\mathit{x}\mathbf{-}\mathbf{1}\mathbf{=}\mathit{y}\mathbf{-}\mathbf{1}$. It follows that $\mathit{x}\mathbf{=}\mathit{y}$, completing the inductive step.

Short answer

The error is in the statement $\max (x-1\mathit{,}y-1)\mathit{=}k$.

Step-by-step solution

Mathematical Induction

The principle of mathematical induction is to prove that $P\left(n\right)$is true for all positive integer n in two steps.

1. Basic step: To verify that $P\left(1\right)$is true.

2. Inductive step: To prove the conditional statement if $P\left(k\right)$is true then $P(k+1)$is true.

Wrong in the proof

The induction hypothesis is true for only positive integers. For x=1 and y=2 in the statement$\max (x-1,y-1)=k$. The value of x-1 is zero which is not a positive integer.

So, the induction hypothesis is not applicable.

Therefore, the statement $\max (x-1,y-1)=k$is wrong.