Question

) Let R be the relation on R×R defined by xRy if and only if x−y is an integer. Prove R is an equivalence relation and describe the partition of R×R determined by R.

Answer 1

Hi,

Explanation:

`xRy\ \Longleftrightarrow\ x-y\ \in\ \mathbb{Z}\\\\1. \ reflexive:\\xRx\ \Longleftrightarrow\ x-x=0\ \in\ \mathbb{Z}\\\\\\2.\ symetric:\\xRy\ \Longleftrightarrow\ x-y\ \in\ \mathbb{Z}\Longleftrightarrow\ y-x=-(x-y)\ \in\ \mathbb{Z}\\3. transitive:\\\\\left\{\begin{array}{ccc} xRy \Longrightarrow \ x=y+n_1, \ n_1\in\ \mathbb{Z}\\yRz \Longrightarrow \ y=z+n_2, \ n_2\in\ \mathbb{Z}\\x=y+n_1=z+n_2+n1=z+n_3 \Longrightarrow \ xRz,&n_3=n_1+n_2 \in\ \mathbb{Z}\\\end{array}\right.`

The partition is all the lines parallel to y=x passing trough (n,0) n being an integer.