Hey everyone,
Today’s problem is a proof derived from mathematical fallacy. See if you can spot the error! (Hint: you’re not Chuck Norris).
Problem 7:
Find the error in the following “proof.” Let x = y. Then:
$latex displaystyle begin{aligned} x^2&=xy,\ x^2-y^2&= xy-y^2,\ (x+y)(x-y)&= y(x-y),\ x+y&=y,\ 2y&=y,\ 2&=1. end{aligned}$
Solution 7:
We know that something is obviously wrong with this proof since we know that 2 ≠ 1. It took me a while to solve this, but after a few minutes of staring at the problem I think I’ve found the solution. Consider the following lines:
$latex displaystyle (x+y)(x-y) = y(x-y),$
$latex displaystyle x+y=y$
Here, the proof writer attempts to divide by zero. In other words:
$latex displaystyle (x+y)(x-y)(x-y)^{-1} = y(x-y)(x-y)^{-1}$
However, x = y, x – y = 0, so:
$latex displaystyle (x-y)^{-1}=text{undefined}.$
I don’t have a solutions manual for these problems, so feel free to disagree if you’re not convinced. I am, however, quite certain that this is a correct solution.
Note: I used a new command today, the begin{aligned} environment to align the equal signs in my first box of $latex LaTeX$ equations, and the target=”blank” attribute to open up links in a new window.