hby0214
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.



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helder_edwin
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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 \(z^4=x^2\).

helder_edwin
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do u remember the definition of function?

hby0214
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A function is a rule that assigns a unique element in Rn to Rm. It is one to one.

helder_edwin
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the last part is something else.

helder_edwin
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a function assigns a UNIQUE element to EVERY element of its domain.

hby0214
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Ok I understand that.

hby0214
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So a certain w and x cannot have two outputs.

helder_edwin
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what does (w,x)=(1,1) get assigned to?

helder_edwin
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yes. precisely.

hby0214
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(1,1) is assigned to (1,1)?

helder_edwin
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just that?

helder_edwin
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let's see:
\[ \large y=w=1 \]
right?

hby0214
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right

helder_edwin
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BUT
\[ \large z^4=x^2=(1)^2=1\Rightarrow z=\pm\sqrt[4]{1}=\pm1 \]

helder_edwin
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so
\[ \large (1,1)\mapsto(1,1) \]
and
\[ \large (1,1)\mapsto(1,1) \]

hby0214
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So does that mean T is not a function?

helder_edwin
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yes. that's what u were asked to prove.

hby0214
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T is graph of something, but that something is not a function. right?

helder_edwin
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yes.

hby0214
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Thank you SO much!

helder_edwin
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u r welcome.