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hby0214 Group Title

Show that the set T ={(w,x,y,z)∈R4 such that y=w and x^2 =z^4} is not the graph of any function of w and x.

  • 2 years ago
  • 2 years ago

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  1. helder_edwin Group Title
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    well u have the correspondence (i don't know if this is the word in english) \[ \large (w,x)\mapsto(y,z) \] such that \(y=w\) and \(z^4=x^2\).

    • 2 years ago
  2. helder_edwin Group Title
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    do u remember the definition of function?

    • 2 years ago
  3. hby0214 Group Title
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    A function is a rule that assigns a unique element in Rn to Rm. It is one to one.

    • 2 years ago
  4. helder_edwin Group Title
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    the last part is something else.

    • 2 years ago
  5. helder_edwin Group Title
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    a function assigns a UNIQUE element to EVERY element of its domain.

    • 2 years ago
  6. hby0214 Group Title
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    Ok I understand that.

    • 2 years ago
  7. hby0214 Group Title
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    So a certain w and x cannot have two outputs.

    • 2 years ago
  8. helder_edwin Group Title
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    what does (w,x)=(1,-1) get assigned to?

    • 2 years ago
  9. helder_edwin Group Title
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    yes. precisely.

    • 2 years ago
  10. hby0214 Group Title
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    (1,-1) is assigned to (1,1)?

    • 2 years ago
  11. helder_edwin Group Title
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    just that?

    • 2 years ago
  12. helder_edwin Group Title
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    let's see: \[ \large y=w=1 \] right?

    • 2 years ago
  13. hby0214 Group Title
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    right

    • 2 years ago
  14. helder_edwin Group Title
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    BUT \[ \large z^4=x^2=(-1)^2=1\Rightarrow z=\pm\sqrt[4]{1}=\pm1 \]

    • 2 years ago
  15. helder_edwin Group Title
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    so \[ \large (1,-1)\mapsto(1,1) \] and \[ \large (1,-1)\mapsto(1,-1) \]

    • 2 years ago
  16. hby0214 Group Title
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    So does that mean T is not a function?

    • 2 years ago
  17. helder_edwin Group Title
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    yes. that's what u were asked to prove.

    • 2 years ago
  18. hby0214 Group Title
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    T is graph of something, but that something is not a function. right?

    • 2 years ago
  19. helder_edwin Group Title
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    yes.

    • 2 years ago
  20. hby0214 Group Title
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    Thank you SO much!

    • 2 years ago
  21. helder_edwin Group Title
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    u r welcome.

    • 2 years ago
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