I answered this wrong on a test, but now I've come up with a different solution.(adsbygoogle = window.adsbygoogle || []).push({});

Problem: Prove that a relation [itex]xRy\Leftrightarrow x-y\in\mathbb{Z}[/itex] defined on [itex]\mathbb{R}[/itex] is an equivalence relation.

Solution:

1.) Reflexivity: [itex]xRx,\forall x\in\mathbb{R}[/itex]

For every [itex]x[/itex] we have [itex]x-x=0[/itex] which is an integer, so reflexivity holds.

2.) Symmetricity: [itex]xRy\Rightarrow yRx,\forall x,y\in\mathbb{R}[/itex]

If for all [itex]x,y\in\mathbb{R}[/itex] we have [itex](x-y)\in\mathbb{Z}[/itex], then [itex]y-x=-1\cdot(x-y)[/itex] (any integer multiplied by -1 is also an integer) and thus [itex](y-x)\in\mathbb{Z}[/itex] and the relation is symmetric.

3.) Transitivity: [itex]xRy\wedge yRz\Rightarrow xRz,\forall x,y,z\in\mathbb{R}[/itex]

For some [itex]x,y,z\in\mathbb{R}[/itex] we have [itex](x-y)+(y-z)=x-y+y-z=x-z[/itex] (the sum of two integers is also an integer) and thus [itex](x-z)\in\mathbb{Z}[/itex]. The relation is also transitive.

Is this it???

PS: sorry for the poor spelling

- Kamataat

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A proof

**Physics Forums | Science Articles, Homework Help, Discussion**