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Homework Help: Equivalence relations

  1. Mar 6, 2010 #1
    1. The problem statement, all variables and given/known data

    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.


    2. Relevant equations


    3. 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.
     
  2. jcsd
  3. Mar 6, 2010 #2

    vela

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    Yes, that's correct.

    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.
     
  4. Mar 6, 2010 #3
    Ok, then I think it is an equivalence relation. Would the equivalence classes be [v] = {av} where a is in R?
     
  5. Mar 6, 2010 #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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