Proof of the Simple Theorem: x^2 >= 0 for Real x

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Discussion Overview

The discussion centers around the proof of the theorem stating that for any real number x, x² is greater than or equal to zero. Participants examine the validity of a proposed proof and the reasoning behind its incorrectness, focusing on logical implications and proof techniques.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant presents a proof that claims x² is non-negative by assuming the opposite and deriving a contradiction using a specific example (x=3).
  • Another participant questions the clarity of the brain teaser and seeks to understand the nature of the proof's incorrectness.
  • A third participant reiterates the confusion regarding the proof's validity and emphasizes the need for a general argument rather than relying on a single example.
  • A later reply critiques the logical structure of the proof, arguing that the assumption made does not properly lead to the conclusion required for a valid reductio ad absurdum argument.

Areas of Agreement / Disagreement

Participants express disagreement regarding the validity of the proof presented. There is no consensus on the correctness of the proof or the reasoning behind its flaws.

Contextual Notes

Participants highlight the need for a general proof that applies to all real numbers rather than relying on specific instances, indicating a limitation in the original proof's approach.

Who May Find This Useful

Individuals interested in mathematical proofs, logic, and the foundations of real analysis may find this discussion relevant.

Howers
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Theorem: Suppose that x is real, then x^2 > or = 0.
Incorrect Proof: Suppose not. Then x is real and x^2<0. Consider x=3. 9<0, a contradiction. Therefore x^2 >=0.
 
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I'm sorry, but what exactly is the brain teaser at hand here?
 
Werg22 said:
I'm sorry, but what exactly is the brain teaser at hand here?
Why the proof is incorrect, even though seemingly it looks like it is.
 
I don't see how could this fool anyone... Given the assumption P -> -Q , --Q(3) does not give rise to P -> Q. One needs to show that the assumption P -> -Q gives rise to --Q (a statement for all x being considered, not just 3) and therefore conclude P -> Q by reductio ad absurdum.
 

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