Equivalence relations and addition

  1. 1. The problem statement, all variables and given/known data

    prove that if a~a' then a+b ~ a' + b

    2. Relevant equations



    3. The attempt at a solution

    I can prove that if a=a' then a+b = a' + b but how can I apply this to any equivalence relation
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,783
    Staff Emeritus
    Science Advisor

    Your question makes no sense at all. An equivalence relation can be established on any set whatsoever- I could, for example, say that two automobiles are equivalent if and only if they were manufactured by the same company- so "a+ b" makes no sense in general.

    Further, even if we assume that you are talking about numbers, whether it is true that a+ b= a'+ b', depends upon exactly what the equivalence relation is! It is NOT true for any equivalence relation on numbers. I can, for example, define a~ b if and only if |a|= |b|. I can then take a= 5, a'= -5, b= 4, b'= 4. It is NOT true that a+ b= 5+ 4= 9 is equal to a'+ b'= -5+ 4= -1.
     
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