Can y^2 = x^2 be parametrized?

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Discussion Overview

The discussion revolves around the possibility of parameterizing the equation y² = x². Participants explore whether this can be achieved through continuous or non-continuous curves, considering the implications of connectedness and cardinality in their arguments.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the graph of y² = x² resembles the lines y = x and y = -x, questioning the possibility of parameterization.
  • Another participant argues that if parameterization requires a continuous curve, it is not possible due to the nature of the graph when the origin is removed, leading to four disconnected pieces.
  • There is a proposal that if non-continuous parameterization is allowed, it could be achieved through a cardinality argument.
  • One participant humorously suggests a simple parameterization with x² = t and y² = t, but later clarifies that they are looking for a function of t for both x and y.
  • A participant expresses uncertainty about explicit parameterizations but notes that any proper real interval has the same cardinality as y² = x², implying the existence of an onto map from an interval to the equation.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method of parameterization, with some agreeing on the impossibility of continuous parameterization while others explore non-continuous options. The discussion remains unresolved regarding explicit parameterizations.

Contextual Notes

Participants acknowledge limitations in their arguments, particularly regarding the nature of connectedness and the implications of cardinality without providing specific parameterizations.

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The graph of [itex]y^2 = x^2[/itex] (1) looks simply like [itex]y = x[/itex] (2) and [itex]y = -x[/itex] (3) plotted on the same axis.

Is it possible to parameterize (1)?
 
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Parametrized by a continuous curve? If so, no, because removing the origin from y^2 = x^2 gives 4 pieces, whereas removing a point from an interval (a, b) gives 2 pieces. More precisely, the image of (a,c) U (c, b) where a < c < b is going to be the union of two connected sets, and this can't possibly be 4 non-intersecting connected pieces.

If not continuous, then yes, simply by a cardinality argument.
 
Thanks Werg22! I meant parametrized by expressing y in terms of t and x in terms of t. I understand why it can't be a continuous curve. How would you parametrize it, not continuously, "by a cardinality argument"?
 
This may be silly, but:
x2=t and y2=t
 
mathman said:
This may be silly, but:
x2=t and y2=t

Hahaha, I knew that :-p. I meant, x = some function of t and y = some function of t. And I don't think [itex]x = \pm \sqrt{t}[/itex] and [itex]y = \pm \sqrt{t}[/itex] counts.
 
Unit said:
Thanks Werg22! I meant parametrized by expressing y in terms of t and x in terms of t. I understand why it can't be a continuous curve. How would you parametrize it, not continuously, "by a cardinality argument"?

I don't know of any explicit parametrization, but any proper real interval has the same cardinality as y^2 = x^2, so an onto map from (a,b) to y^2 = x^2 exists. This is simply an existential statement, take it for what it's worth.
 

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