Question #105102
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
1
Expert's answer
2020-03-13T09:48:42-0400

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

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

Hence \sim is symmetric.

Transitive: If a,b,cR and abZ,bcZ then ab=na,b,c\in R \space and \space a-b\in Z,b-c\in Z \space then \space a-b=n for some nZ and bc=mn\in Z \space and \space b-c=m for some mZ then a=n+b and c=bm    ac=(n+b)(bm)=nmZ.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.


[5\sqrt{5}]== { xR:x5x\in R :x\sim \sqrt{5} }

== { xR:x5Z i,e x5=nx\in R:x-\sqrt{5} \in Z \space i,e \space x-\sqrt{5}=n for some nZn\in Z }

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS