1. ## Show Equivalence Relation

Let T={(x,y,z) in R^3 | (x,y,z) ≠ (0,0,0)}. Define ~ on T by (x1, y1, z1) ~ (x2, y2, z2) if there exists a nonzero real number β such that x1 = βx2, y1 = βy2 and z1 = βz2. Show that ~ is an equivalence relation on T.

I know to prove equivalence relation you need to prove that reflexive, symmetric and transative all exists. I am not sure how to prove them. Any help/suggestions would be great.

2. Originally Posted by page929
Let T={(x,y,z) in R^3 | (x,y,z) ≠ (0,0,0)}. Define ~ on T by (x1, y1, z1) ~ (x2, y2, z2) if there exists a nonzero real number β such that x1 = βx2, y1 = βy2 and z1 = βz2. Show that ~ is an equivalence relation on T.

I know to prove equivalence relation you need to prove that reflexive, symmetric and transative all exists. I am not sure how to prove them. Any help/suggestions would be great.

Ok, what have you done? Reflexivity and symmetry are almost immediate: what about transitivity?

Tonio

3. I am not even sure on how to begin reflexivity. I am hoping that with some help on this I will be able to work through the other two.

4. Originally Posted by page929
I am not even sure on how to begin reflexivity.
Do you know the definition of reflexivity?
If not, then it is pointless to even try this problem.
If you do understand what reflexive means then you know $\beta =1$ works.