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Math Help - Partition and Equivalence class

  1. #1
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    Partition and Equivalence class

    Define the relation R on the real numbers R by xRy if and only if x - y belong to Z, that is, x - y is an integer.
    a) Describe the Partition.
    (c) Prove that for every x E R there exists a y E [0; 1) such that x Ey/R.
    (note:For any real number a and nonzero real
    number b, there exists an integer q and a real number r such that a = qb+r and 0 <= r < lbl.)

    I am having a hard time with partition, i don't know how to describe them explicitly
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  2. #2
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    Quote Originally Posted by mystic View Post
    Define the relation R on the real numbers R by xRy if and only if x - y belong to Z, that is, x - y is an integer.
    a) Describe the Partition.
    (c) Prove that for every x E R there exists a y E [0; 1) such that x Ey/R.
    You must know the floor function (also known as the greatest integer).
    \left( {\forall x \in \mathbb{R}} \right)\left[ {\left\lfloor x \right\rfloor  \in \mathbb{Z} \wedge \left\lfloor x \right\rfloor  \leqslant x < \left\lfloor x \right\rfloor  + 1} \right]

    That means 0 \leqslant x - \left\lfloor x \right\rfloor  < 1\text{ or }x - \left\lfloor x \right\rfloor  \in [0,1)

    Thus each element in [0,1) determines an equivalence class.
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