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Math Help - Proving equivalence classes bijective to the set of points on the unit circle

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    Proving equivalence classes bijective to the set of points on the unit circle

    Define a relation on R as follows. Two real numbers x, y are
    equivalent if x - y  \epsilon Z . Show that the set of equivalence classes of this relation is bijective to the set of points on the unit circle.

    A part of the problem that I've omitted asked us to prove that the relation is an equivalence one -- I've done that. I've also defined the set of points on the unit circle, which is \{ a,b \epsilon R | \sqrt{x^{2}+y^{2}} \} I don't know where to go from here, though.
    Last edited by Plato; November 1st 2009 at 01:19 PM. Reason: clean up
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    Quote Originally Posted by ams2990 View Post
    Define a relation on R as follows. Two real numbers x, y are
    equivalent if x - y  \epsilon Z . Show that the set of equivalence classes of this relation is bijective to the set of points on the unit circle.

    A part of the problem that I've omitted asked us to prove that the relation is an equivalence one -- I've done that. I've also defined the set of points on the unit circle, which is \{ a,b \epsilon R | \sqrt{x^{2}+y^{2}} \}
    Here are some observations.
    \Phi (t) = \left( {\cos (2\pi t),\sin (2\pi t)} \right) is a bijection of [0,1) to the unit circle.

    Using the floor function, \left( {\forall r \in R} \right)\left[ {r - \left\lfloor r \right\rfloor  \in [0,1)} \right]

    But you should see that \left( {\forall r \in R} \right)\left[ {r - \left\lfloor r \right\rfloor } \right] is an equivalence class for this equivalence relation.
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