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.

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