# Confusion about division by zero in sets

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• Andrew Wright
In summary, the confusion lies in the fact that division by zero is undefined, but the point (0,0) appears in the set of values where x=y. However, this point does not appear in the set of values where 1=y/x, as the transformation from x=y to x/y=1 is not equivalent. This raises the question of whether the sets of points where x=y and where 1=y/x are the same, and the answer is no due to the exclusion of y=0 in the transformation. The answer to this question may also depend on the assumptions made about division by zero, and it is referred to as a "thing" in mathematics.
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?

Andrew Wright said:
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.

$$x=y \nLeftrightarrow \dfrac{x}{y}=1$$
Andrew Wright said:
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.
Andrew Wright said:
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.

Thanks, sufficient for me.

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