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Proofs Question: Equivalence Relation and Classes

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data

    Define a relation ~ on ℝ by

    a~b if and only if a-b∈Q.

    i) Show that ~ is an equivalence relation.

    ii) Show that

    [a]+=[a+b]

    is a well-defined addition on the set of equivalence classes.

    2. Relevant equations

    Q represents the set of rational numbers.
    An Equivalence Relation must be Reflexive (a~a), Symmetric (a~b implies b~a), and Transitive (a~b and b~c implies a~c).

    3. The attempt at a solution

    So this is what I came up with and for the most part I think I'm right. If anyone notices anything wrong or needs more information, please let me know and I'll do what I can.

    i) Reflexive: a~a → a-a=0 → 0∈Q

    Symmetric: a~b → b~a

    Here I just showed how if a-b∈Q, then so is it's negative. Thus leading to b~a.

    a~b → a-b∈Q → -(a-b)∈Q → -a+b∈Q → b-a∈Q → b~a

    Transitive: a~b and b~c → a~c

    Here I showed that a-b∈Q and b-c∈Q added together will give a-c∈Q. Showing that a~c.

    a-b∈Q and b-c∈Q

    (a-b)+(b-c)∈Q → a-b+b-c∈Q → a-c∈Q → a~c

    ii) [a]={c1 | a~c1 <-> a-c1∈Q}
    ={c2 | b~c2 <-> b-c2∈Q}

    [a]+=[a+b]
    (a-c1)+(b-c2)=(a+b)-(c1+c2)∈Q

    [Side note: (a+b)∈ ℝ and (c1+c2)∈ ℝ ]
     
    Last edited: Nov 7, 2011
  2. jcsd
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