Be careful with the equivalence classes.
.
Which of the following relation on the given sets (S) define equivalence relations? Describe the equivalence classes where appropriate:
a) S= R3
x~y if there is a 1 dimensional subspace of R3 which contains x and y
b) S= R3 - {0}
x~y if there is a 1 dimensional subspace of R3 which contains x and y
c) S = R3
x~y if there is a 2D subspace of R3 which contains x and y
My Answers:
I always get confused with equivalence relations so I just wanted to give my thoughts to see if I am on the right lines!
a) I think this is an equivalence relation as it is reflexive, symmetric and transitive.
I think the equivalence classes would be the individual lines that the two points lie on. However this would mean that there would be infinitely many equivalence classes so seems a bit suspect?
b) I dont think this relation is an equivalence relation as it is not transitive. I looked at the case of a triangle with two of the lines going through the origin, but the other therefore wouldnt.
c) I assumed this question would be similar to a. Reflexive, Symmetric and Transitive, with the equivalence classes being the individual planes that the lines lie within.
[/b]
I needed to edit this post, because I was in a hurry last time, and overlooked the zero vector. Here is the corrected solution:
First of all, I will be assuming that your "if"s are "iff"s, and that "subspace" refers to a vector subspace. That said...
(a) Since we can choose with (for example ), then , and so transitivity does not hold. So is not an equivalence relation.
(b) Reflexivity and symmetry are obvious here. For transitivity, recall that iff for some (the underlying field). Suppose . Then there are with , which means . So transitivity holds, and is an equivalence relation.
(c) Transitivity is again the issue, here, but this time it might work even with the zero vector. Indeed, for any , we have , where is a two-dimensional vector space. So , and transitivity holds such that is an equivalence relation (and ).
First, who would say this instead of something like "the line segment (x, y) is a subset of S"?a) S= R3
x~y if there is a 1 dimensional subspace of R3 which contains x and y
Anyway, I agree that (a) and (c) are equivalence relations and (b) is not.
I am not sure about this. E.g., (0,0,1) ~ (0,1,0) because they lie on the same line, but there is no such a that (0,0,1) = a * (0,1,0).
The relation ~ in (b) is not empty; it is already defined and, for example, (0,0,1) ~ (0,0,2).
Concerning equivalence classes, in (a) it's not lines because any three points are equivalent, and similarly for (c).