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The set of solutions for $y=x+1$ can be represented by a set of ordered pairs of the form $(y,x)$ or $(x,y)$ which is obviously always used. So these 2 sets define different subsets of $\mathbb R^2$, but i would graph them the same on the $x-y$ plane.

I'm confused because my notes say the element $(a,b)$ of $\mathbb R^2$ is represented by the point in the Cartesian plane whose $x$-coordinate is $a$ and $y$ coordinate is $b$, so where am I wrong?

thanks

Mauro ALLEGRANZA
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Carlos Bacca
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  • Because $y$ is function of $x$; see definition : $y=x+1$ and we graph a function $f$ as a set of points : $(x,f(x))$. – Mauro ALLEGRANZA Sep 06 '18 at 14:43
  • thanks sot hats the convention when y=f(x) to represent the solutions as an ordered pair of the form (x,y) but if y is not a function of x, then can the solutions be of the form (x,y) or (y,x) which would lead to different subsets but the same graphical representation – Carlos Bacca Sep 06 '18 at 14:55
  • I think that If y=x+1 then it can define two different functions on whether you consider the solutions the form (x,y) or (y,x), either y is a function of x or x is a function of y.when displaying the graph of a function the variable representing the second component of the ordered pair, or the dependant variable, is on the vertical axis so the two sets would actually have a different graphical representation (because they are different functions) – Carlos Bacca Sep 06 '18 at 15:34
  • if y is not a function of x or x is not a function of y then the set of ordered pairs representing the solutions can be of the form (x,y) or (y,x) these sets can be plotted on the x-y plane and give the same representation because they are solution from the same equation. The last point i made contradicts the definition in my question – Carlos Bacca Sep 06 '18 at 15:35

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