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
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.