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
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Expert's answer
2020-03-13T09:48:42-0400
Reflexive: if a∈R then a−a=0∈Z . Hence ∼ is reflexive.
Symmetric: if a,b∈Randa−b∈Z then b−a also belongs to Z
Hence ∼ is symmetric.
Transitive: If a,b,c∈Randa−b∈Z,b−c∈Zthena−b=n for some n∈Zandb−c=m for some m∈Zthena=n+bandc=b−m⟹a−c=(n+b)−(b−m)=n−m∈Z.
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