Domain: why is domain of f(x, y) given as "R^2" rather than just "R"?

SB_

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Domain: why is domain of f(x, y) given as "R^2" rather than just "R"?

Hello all, I have a quick question I was hoping somebody couldclarify.
If I have a function of two variables f(x,y), why is thedomain of the function “any real number” squared rather than just any realnumber?
Is it just because the inputs are coming from both x and y?
Thanks in advance.
 

pka

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Hello all, I have a quick question I was hoping somebody could clarify.
If I have a function of two variables f(x,y), why is the domain of the function “any real number” squared rather than just any real number? Is it just because the inputs are coming from both x and y?
I can give a very technical set-theoretic answer. Someone else may simplify this answer.
The function \(\displaystyle f(x,y)=x^2+xy+y^2\) is a mapping \(\displaystyle (\Re\times\Re)\to\Re\).
Now \(\displaystyle (\Re\times\Re)=\{(x,y):~\{x,y\}\subset\Re\}\), ordered pairs of reals. The cross product of the reals with the reals is the domain.
Thus in a loose sense the domain of the function could be referred to as the real numbers.
 

Ishuda

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Hello all, I have a quick question I was hoping somebody couldclarify.
If I have a function of two variables f(x,y), why is thedomain of the function “any real number” squared rather than just any realnumber?
Is it just because the inputs are coming from both x and y?
Thanks in advance.
As pka said, formally the domain is \(\displaystyle (\Re\times\Re)=\{(x,y):~\{x,y\}\subset\Re\}=\{(x,y):~\, x\, and\, y\, \epsilon\, \Re\}\)

Informally, \(\displaystyle (\Re\times\Re)\) is often referred to as R2. Sometimes the informal notation is made formal by defining R2 as \(\displaystyle (\Re\times\Re)\). This is true of the higher order spaces/domains/dimensions also, i.e. R3 as \(\displaystyle (\Re\times\Re\times\Re)\), etc.
 

Prove It

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Hello all, I have a quick question I was hoping somebody couldclarify.
If I have a function of two variables f(x,y), why is thedomain of the function “any real number” squared rather than just any realnumber?
Is it just because the inputs are coming from both x and y?
Thanks in advance.
Every independent variable has its own domain, but to show the complete domain of a multivariable function is to show the domain of each variable. Because each variable is its own "dimension", to show them altogether you would use the "times" notation.

Thus the domain of \(\displaystyle f\left( x, y, z \right) \) for example would be \(\displaystyle \textrm{Dom}\left( x \right) \times \textrm{Dom} \left( y \right) \times \textrm{Dom} \left( z \right)\).

If the domains happen to be the same, then we can use a power notation, the same way we would with multiplying the same amount by itself. So if the domain of x, y, z were all R, then we could write \(\displaystyle \mathbf{R} \times \mathbf{R} \times \mathbf{R} = \mathbf{R}^3\).
 
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stapel

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$ \textrm{Dom}\left( x \right) \times \textrm{Dom} \left( y \right) \times \textrm{Dom} \left( z \right)$.

$ \mathbf{R} \times \mathbf{R} \times \mathbf{R} = \mathbf{R}^3$.
Are these lines displaying as properly formatted on your screen? Because I'm seeing lots of dollar-signs, back-slashes, unfamiliar commands, etc, rather than mathematical text. :oops:
 

pka

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Every independent variable has its own domain, but to show the complete domain of a multivariable function is to show the domain of each variable. Because each variable is its own "dimension", to show them altogether you would use the "times" notation.

Thus the domain of \(\displaystyle f\left( x, y, z \right) \) for example would be \(\displaystyle \textrm{Dom}\left( x \right) \times \textrm{Dom} \left( y \right) \times \textrm{Dom} \left( z \right)\).

If the domains happen to be the same, then we can use a power notation, the same way we would with multiplying the same amount by itself. So if the domain of x, y, z were all R, then we could write \(\displaystyle \mathbf{R} \times \mathbf{R} \times \mathbf{R} = \mathbf{R}^3\).
Wrap simple LaTeX in [ tex][/tex ] without the first and last space.
 

Prove It

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I apologise, every other mathematics forum I am on uses dollar signs to tell the forum to compile the LaTeX. I also assumed that the mod would edit it as they review my posts beforehand. Thank you for telling me how to do it right :)
 
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