Discrete fallacious proof

1. Sep 29, 2008

Enjoicube

1. The problem statement, all variables and given/known data
Alright here it is:

Theorem: if there exists an x belonging to reals such that (x^2)-x-2=(x^2)-4 then 1=2.

Remark: note that there is such an x belonging to reals.

Proof:

1) by hypothesis assume there exists an X belonging to reals such that (x^2)-x-2=(x^2)-4

2)factor each side,

3)resulting in (x-2)(x+1)=(x-2)(x+2)

4)divide each side by (x-2),

5)resulting in x+1=x+2

6)subtract x from each side, resulting in 1=2

1) What terminology (quantifiers, predicates) can be used to express the entire statements 1,3,5

2)Why is this proof fallacious, refer to statements by their numbers. Hint: What are the domains for each statement?

2. Relevant equations

3. The attempt at a solution

Ok, my attempt at part one (is
Theorem: ($$\exists$$X $$\in$$$$\Re$$) $$\right arrow$$ ((x^2)-x-2=(x^2)-4))

Statement 1: $$\exists$$X $$\in$$$$\Re$$ ((x^2)-x-2=(x^2)-4))

Statement 3:$$\exists$$X $$\in$$$$\Re$$(x-2)(x+1)=(x-2)(x+2)

Statement 5:$$\exists$$X $$\in$$$$\Re$$x+1=x+2

For part 2, i am quite lost, the problem is the use of existential quantifier. My guess is a division by zero somewhere, but otherwise, I need a bigger hint.
1. The problem statement, all variables and given/known data
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Sep 29, 2008

HallsofIvy

Staff Emeritus
The theorem starts "there exists an X belonging to reals such that (x^2)-x-2=(x^2)-4". Okay, is that true? If not then the conclusion is false because the hypothesis is false! If it is true, what is that x? In other words, solve x^2- x- 2= x^2- 4. I will tell you right now that the hypothesis is true but once you have determined what that x is, you will see why dividing both sides of the equation by x- 2 is an error.

3. Sep 29, 2008

Enjoicube

Aha! got it. Thank you so much for that. In retrospect I really should have noticed this x value.

4. Sep 29, 2008

HallsofIvy

Staff Emeritus
In retrospect, I should have been a genius!

5. Sep 30, 2008

the problem with this "proof" is simply that dividing by $$x-2$$ is division by zero, since 2 is a solution to $$x^2 - x -2 = x^4 - 4$$