Confusion about division by zero in sets

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.
  • #1
Andrew Wright
120
19
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?
 
Physics news on Phys.org
  • #2
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.
 
  • #3
Thanks, sufficient for me.
 

Similar threads

  • General Math
2
Replies
47
Views
4K
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
621
  • Set Theory, Logic, Probability, Statistics
Replies
2
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
752
  • Set Theory, Logic, Probability, Statistics
Replies
13
Views
993
  • Set Theory, Logic, Probability, Statistics
Replies
5
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
18
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
3
Views
930
Back
Top