B Confusion about division by zero in sets

AI Thread Summary
Division by zero is often considered undefined, leading to confusion about the relationship between the sets defined by x=y and 1=y/x. The point (0,0) is included in the set where x=y but not in the set where 1=y/x, as the transformation from x=y to x/y=1 is not equivalent when y=0. This discrepancy arises because dividing by y implicitly excludes the case where y equals zero. The discussion highlights the importance of understanding equivalence transformations in mathematical operations. Ultimately, the sets are not identical due to the restrictions imposed by division by zero.
Andrew Wright
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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?
 
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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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