$f) x^2-5x+6=y^2-5y+6$

$g) x^2+y^2=1$

Decide whether or not it’s a reflexive, symmetric, transitive and equivalence relation. If R is an equivalence relation, describe the equivalence classes.

I guess the first one is an equivalence. But I'm having trouble what are the equivalence classes. For the second one, it's not reflexive since it's not for all x in Z? It's symmetric but not transitive?


First one is an equivalence relation, and you can easily prove it. For the first one, equivalence classes are the set of points where the function takes the same value. For example, roots of the polynomial, {2,3} will form an equivalence class ([2]=[3]).

For the second one, it is not reflexive or transitive, but symmetric as you inferred.

  • $\begingroup$ ^ what if the first was like (x-1)^2 = (y-1)^2. Do we say [1] is the equivalence class? $\endgroup$ – user228783 Apr 13 '15 at 4:09
  • $\begingroup$ Please note: [1] denotes the equivalence class of 1, and in that case, [1]={1}. In first question above, [2]=[3]={2,3} $\endgroup$ – Jesse P Francis Apr 13 '15 at 4:21

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