Answer to Question #105102 in Abstract Algebra for Sourav Mondal

Question

Define ~ on R by ‘ a ~ b iff a − b∈Z ’. Check whether or not ~ is an equivalence relation on R. If it is, find [√5 ]Else, give another equivalence relation on R

Answer 1

Reflexive: if $a \in R$ then $a-a=0\in Z$ . Hence $\sim$ is reflexive.

Symmetric: if $a,b\in R \space and \space a-b\in Z$ then $b-a$ also belongs to Z

Hence $\sim$ is symmetric.

Transitive: If $a,b,c\in R \space and \space a-b\in Z,b-c\in Z \space then \space a-b=n$ for some $n\in Z \space and \space b-c=m$ for some $m\in Z\space then \space a=n+b \space and \space c=b-m \implies a-c=(n+b)-(b-m)=n-m\in Z .$

Therefore $\sim$ is transitive.

Hence $\sim$ is an equivalence relation on R.

[$\sqrt{5}$]$=$ { $x\in R :x\sim \sqrt{5}$ }

$=$ { $x\in R:x-\sqrt{5} \in Z \space i,e \space x-\sqrt{5}=n$ for some $n\in Z$ }

$=$ { $x\in R: x=n+\sqrt{5}$ where $n\in Z$ }