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Can we think of plane in three-dimensional space as a function of 2 variables? In other words, is the plane $z=f(x,y)=x+y$ a function?

ssane
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    It’s the graph of a function. – amd Jan 12 '20 at 10:17
  • Thanks! I susect that plane is a graph of the function $f(x,y)=x+y$. Then, is a hyperplane also a graph of a function? – ssane Jan 12 '20 at 10:19
  • Note, though, that just as a line parallel to the $y$-axis in $\mathbb R^2$ isn’t graphs of any function of $x$, neither is a plane parallel to the $z$-axis the graph of any function of $x$ and $y$. – amd Jan 12 '20 at 10:21
  • Sorry, I didn't get your last comment. So, plane is a graph of the function $z=x+y$? – ssane Jan 12 '20 at 10:24
  • You can maybe ask: Can a plane/hyperplane be described by a function? And well, the answer is a yes, it can! And I guess you also figured out how to do that in 2d, going to higher dimensions isn't that much more difficult :) – Imago Jan 12 '20 at 10:37
  • @Imago Okay, I see. If I ask the reverse question: can be a graph of a function be a plane, then what is the answer? (see please jjacquellin answer: No function is a plane). – ssane Jan 12 '20 at 10:44

1 Answers1

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A plane is not a function.

No function is a plane.

Both are of different nature. A function is a relation that uniquely associates members of one set with members of another set (multivariate function in the present case). A plane is a geometrical object, a flat two-dimensional surface.

In a given system of coordinates, a plane can be mathematically described by a function which is loosely called "the equation of the plane" relatively to this system of coordinates.

In a given system of coordinates the "equation" of a plane is a function. In a different system of coordinates the "equation" of the same plane is a different function.

JJacquelin
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  • wrong. a binary operation is a function in 3d based on inputs of two of it's coordinates. –  Jan 12 '20 at 10:40
  • Thanks for the answer. You say that no function is a plane. Do you mean that no function's graph has a shape of plane? – ssane Jan 12 '20 at 10:45
  • Bravo for the purists ! Nevertheless, note that the word "operation" is never used in my answer, which is still something else. – JJacquelin Jan 12 '20 at 10:47
  • @sane; Not at all ! No function is a plane but some functions are used to mathematically describe a plane (they "graph has a shape of plane" with you terminology). For example the function $z(x,y)=c_1x+c_2y+c_3$ in Cartesian coordinates or another function is polar coordinates, or others in other systems of coordinates. – JJacquelin Jan 12 '20 at 10:58
  • If I understand right from your answer and comments, plane is a geometrical object and has nothing to do with a function. However, some functions (i.e. $z=f(x,y)=x+y$) has a shape of a plane by some "coincidence", but we can't say that the function is a plane in terms of its geometrical meaning. Right? – ssane Jan 12 '20 at 11:06
  • Yes, for purists it is not correct to say that the function in a plane. But in common use many people loosely confuse both and everybody understand what they mean. – JJacquelin Jan 12 '20 at 11:20
  • I see, thanks!! – ssane Jan 12 '20 at 11:22
  • A plane is a set. – cangrejo Jan 12 '20 at 14:04
  • @broncoAbierto. A plane is a set of points. A function is a relation that associates members of one set with members of another set. The function is a relation. Thus the function is not a set. Thus the function is not the plane. – JJacquelin Jan 12 '20 at 14:27
  • @JJacquelin I agree, but in your answer I am missing this fact, which I think might be helpful to OP. – cangrejo Jan 12 '20 at 14:47
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    @broncoAbierto. Your remark is well-done. I should have more clearly pointed out that in my answer. – JJacquelin Jan 12 '20 at 15:44