# Thread: Two quick questions regarding Partitions and Equivalence Relations

1. ## Two quick questions regarding Partitions and Equivalence Relations

I'm confused on how to do the following problems for my Proofs class (R represents the set of real numbers):

1. For the following equivalence relation, describe the corresponding partition.
Let ~ be the relation on R - {0} given by x ~ y iff xy > 0, for all x,y elements of R - {0}

2. For the following partition, describe the corresponding equivalence relation.
Let D be the partition of R^2 consisting of all circles in R^2 centered at the origin (the origin is considered a "degenerate" circle).

For the second problem, my professor started us off by saying we should write, "Let ~ be the equivalence relation on R^2 given by (x,y) ~ (z,w) iff __________."

My professor gave a few examples but they weren't the same "style" (for lack of a better word) as the homework problems.

And, just to clarify, R - {0} is the non-zero real numbers, right?

2. On the first problem, I started off by having P = {[x] | x is an element of R - {0}} (where [x] is the equivalence class), and I think the next step is to determine which non-zero real numbers y can be multiplied by a non-zero real number x such that xy < 0 ?

3. hint: if x>0 then what is the condition on y such that xy>0? And similarliy what if x<0?

and for 2 - What is the eqiation of a circle centered at (0,0)?

4. Well, if x>0, then y must also be greater than 0, and if x<0, then y<0 ... That's pretty much what I was thinking, I'm just a little confused on how to write it out as a partition?

5. It's been a while since I've used equations of circles, but if I remember correctly, isn't a circle centered at the origin x^2 + y^2 = 0 ?

6. partition 1 - {x|x>0}
partition 2 - {x|x<0}

i think it is as simple as that

7. yes - so what should be relation between (x,y) and (z,w) such that they lie on the same circle?

8. Originally Posted by aman_cc
yes - so what should be relation between (x,y) and (z,w) such that they lie on the same circle?
They would lie on the same circle iff x^2 + y^2 = z^2 + w^2, which implies x=z and y=w. Is that all the problem is asking for?

9. "x^2 + y^2 = z^2 + w^2, which implies x=z and y=w."

This is wrong !! Think about it carefully

10. Originally Posted by aman_cc
"x^2 + y^2 = z^2 + w^2, which implies x=z and y=w."

This is wrong !! Think about it carefully
Originally all I had written down was (x,y) , (z,w) lie on the same circle iff x=z and y=w, but I added in the x^2+y^2 part... is it still wrong?

11. (x,y)~(z,w) if x^2+y^2 = z^2 + w^2

1. Now you should check that above is an equivalence relation
2. Also understand very clearly that x^2+y^2 = z^2 + w^2 does not => that x=z and y=w. And this is the part which is wrong in your argument

12. Originally Posted by aman_cc
(x,y)~(z,w) if x^2+y^2 = z^2 + w^2

1. Now you should check that above is an equivalence relation
2. Also understand very clearly that x^2+y^2 = z^2 + w^2 does not => that x=z and y=w. And this is the part which is wrong in your argument
Ah, yes, I see why I made that mistake now. I think I was trying to use the formula of a circle, but also stuck on the idea that x=z and y=w, but that obviously isn't true because, on the unit circle for example, the points (1,0) and (-1,0) are on the same circle yet 1 does not equal -1. ... Thanks for the help!

13. correct