Well in that case... >=D

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But anyways, the point of the first statement was that you can complete a proof by starting with a statement that you don't know to be true, with a simple subsitution.

Unfortunately that is not true. =( For an example of why it isn't, see my -1 = 1 "proof" above.

The fundamental logic at work here is that if we know X and Y are the same, then for any function f, f(X) and f(Y) must also be the same (that is, X=Y implies f(X)=f(Y)). However the converse is only true for one-to-one functions, those which never "repeat" any value in their range. Taking f(x) = x^2, for example, f(X)=f(Y) does

*not* imply that X=Y; if X and Y are negatives, their squares are still the same.

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Second piece, there are special rules when it comes to sqrt and ^2. You still end up with the same answer of +-X=+-X. Your example doesn't prove anything but in that one example you don't complete your proof by starting with a known false statement.

The only reason there are "special rules" is that x^2 is not one-to-one, and hence it

*has no inverse function*. We use square roots to "undo" squaring, but without additional information it is not possible to tell whether the positive or the negative result is correct. It gets worse with periodic functions -- if sin(x) = 0, there are infinitely many possible values of x, namely all the integer multiples of pi.

My example shows that there is at least one case in which starting with a false statement nevertheless results in something true. Ergo, one cannot assume in general that starting with a statement that may or may not be true will result in a statement reflective of the original's truth value. =3

Anyway, I'm not a big fan of trying to applying rigorous logic to things that are obviously jokes, and I did laugh out loud at your original post. :) I would've kept my mouth shut except that, being a professional mathematician, I was incited to blind rage by your later comment. :P