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Equivalence relations

  • Thread starter Kate2010
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  • #1
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Homework Statement



I have got myself very confused about equivalence relations. I have to determine whether certain relations R are equivalence relations (and if they are describe the partition into equivalence classes, but I'll worry about that once I understand the first part).

Here are some of the relations:

i) S = {x is a real number| x>0}; aRb iff ab = 1

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have several other relations to consider but is someone could give me some ideas with these, hopefully I will be able to transfer my understanding to the others.

Many thanks.


Homework Equations




The Attempt at a Solution



i) S = {x is a real number| x>0}; aRb iff ab = 1

If my understanding is correct, this is not an equivalence relation as aRa is not true for all a in S, e.g. if a = 2 then 2*2=4 is not equal to 1, so R is not reflexive.

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have no idea on this one.
 

Answers and Replies

  • #2
vela
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The Attempt at a Solution



i) S = {x is a real number| x>0}; aRb iff ab = 1

If my understanding is correct, this is not an equivalence relation as aRa is not true for all a in S, e.g. if a = 2 then 2*2=4 is not equal to 1, so R is not reflexive.
Yes, that's correct.

ii) S = R^3\ {0}; vRw iff there is a 1 dimensional subspace of the vector space R^3 which contains v and w.

I have no idea on this one.
Hint: A one-dimensional subspace has a basis containing just one vector, so the subspace is just the set of all multiples of that vector.
 
  • #3
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Ok, then I think it is an equivalence relation. Would the equivalence classes be [v] = {av} where a is in R?
 
  • #4
vela
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Yup, though a can't be 0 since the zero vector isn't in S.
 

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