1. The problem statement, all variables and given/known data For the set ℝ, define ~ as x~y whenever |x| = |y|. Define the addition and multiplication of equivalence classes as: [x]+[y] = [x+y] [x]*[y] = [xy] a) Show that the multiplication of equivalence classes is well defined. b) Give an example that illustrates that the addition of Equivalence classes is not well defined. 2. Relevant equations 3. The attempt at a solution a) [x]*[y]=[xy] Well, let's focus on B for now. b) I don't understand this. How could it not be well defined? This is the definition of addition. I figure we MUST use the fact that x~y means |x| = |y|. So, I think of examples, such as x=1 y=-1 as equivalence relations. And then such examples as x=1/2 and y = -1/2 So we see 1+(-1/2) would be equal to 1/2 while -1 + (1/2) would be equal to -1/2 1/2 does not equal -1/2 but by the definition of equivalence here |1/2| = |-1/2| so it checks out, right? Or am I wrong here and just need to show that 1/2 does not equal -1/2 so we have a contradiction?