Question

Define a relation on ℝ^2 \{(0,0)} by setting (x1, y1) ~ (x2, y2) if there exists a nonzero real number λ for which (x1, y1) = (λx2, λy2). Prove that ~ defines an equivalence relation on ℝ^2 \{(0,0)}. What are the equivalence classes?

Answer 1

@Jemurray3 can u help me?

Answer 2

Can you prove that the relation defined above is an equivalence relation? That should be relatively straightforward. Just prove that it's reflexive, transitive, and symmetric.

Answer 3

okkkkkkk

Answer 4

There are a number of ways you could define the equivalence classes. Do you fully understand what the question is asking?

Answer 5

yes i doooooo

Answer 6

Alright. My suggestion for the equivalence classes are the points on the unit circle, but you can do what you'd like :) Message me if you get stuck.

Answer 7

ya i just did whtvr like it was a stupid discussion question. It doesnt have to be correct i just have to interact and stuff

Answer 8

Well here: The relation says that, with lambda a real number,
\[ (x_1,y_1) = (x_2,y_2) \text{ if } (x_2,y_2) = (\lambda x_1, \lambda y_1) \]
the reflexive property is satisfied with
\[\lambda = 1\]
and 1 is a real.
The transitive property is straightforward to prove, and the symmetric property is satisfied as follows: If
\[(x_2,y_2) = (\lambda x_1, \lambda y_1)\]
then
\[(x_1,y_1) = (\frac{1}{\lambda} x_2, \frac{1}{\lambda} y_1) \]
and if lambda is real, 1/lambda is also real.

Answer 9

Thanks :)