1. Let ~ be the relation on $\mathbb R$ defined by
    a ~ b if and only if |a| = |b|:

(a) Prove that is an equivalence relation.
(b) Give an example of an equivalence class with two elements.
(c) Give an example of an equivalence class with one element.
(d) Give a complete set of equivalence class representatives.

I can do part (a) of the question by showing the symmetry, reflexivity and transitivity but I'm struggling with the rest of the question. Can anyone give me some hints?


Hint: Your equivalence class is $\pm a$ for some number $a$. So give an example where $+a \neq -a$ and give an example where $+a = -a$.

  • $\begingroup$ So for part (b) an equivalence class could be {1,-1} or {2,-2} etc. and part (c) can only be {0} $\endgroup$ – Student101 Jan 20 '15 at 21:05
  • $\begingroup$ Would that make part (c) {0,1,2,3,.......} i.e. all n in Z* $\endgroup$ – Student101 Jan 20 '15 at 21:30
  • $\begingroup$ (b) and (c) are good, but for your representatives why are you choosing only elements from Z? Isn't your equivalence on R? $\endgroup$ – Jim Jan 20 '15 at 23:10
  • $\begingroup$ Ah good point, thanks a lot! $\endgroup$ – Student101 Jan 21 '15 at 15:23

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