Confusion about division by zero in sets

[Andrew Wright]

TL;DR Summary

If you re-arrange x=y to be x/y = 1, do you end up with an identical set after re-arrangement?

So the confusion here is that division by zero is often said to be undefined. So whereas, the point (0,0) certainly appears in the set of values where x=y, does the point (0,0) appear in the set of values where 1=y/x. Why or why not?

In other words are the set of points where x=y the same as the set of points where 1=y/x?

Does the answer depend on what assumptions you start off with about the nature of division by zero? If it is a "thing" who came up with it and what is it called?

 

[fresh_42]

TL;DR Summary: If you re-arrange x=y to be x/y = 1, do you end up with an identical set after re-arrangement?

So the confusion here is that division by zero is often said to be undefined. So whereas, the point (0,0) certainly appears in the set of values where x=y, does the point (0,0) appear in the set of values where 1=y/x. Why or why not?

Because you did not perform an equivalent transformation.

$$
x=y \nLeftrightarrow \dfrac{x}{y}=1
$$

In other words are the set of points where x=y the same as the set of points where 1=y/x?

No, because as you observed, too, ##(x,y)=(0,0)## is a solution on the left but not on the right.

Does the answer depend on what assumptions you start off with about the nature of division by zero? If it is a "thing" who came up with it and what is it called?

It depends on whether you perform equivalence transformations or not. By dividing by ##y## you implicitly ruled out ##y=0##. That's why you lost it.

 

[Andrew Wright]

Thanks, sufficient for me.