# Thread: Partition and Equivalence class

1. ## 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

2. Originally Posted by mystic
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).
$\displaystyle \left( {\forall x \in \mathbb{R}} \right)\left[ {\left\lfloor x \right\rfloor \in \mathbb{Z} \wedge \left\lfloor x \right\rfloor \leqslant x < \left\lfloor x \right\rfloor + 1} \right]$

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

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