Answer to Question #153061 in Abstract Algebra for Girish Chandra

Question #153061
Let s= {p/q €Q|(q,7) =1} Define a relation ~ on s by p/q ~ a/b iff 7| (bp-aq). check whether or not ~ is an equivalence relation on s?
1
Expert's answer
2020-12-29T16:02:20-0500

The relation is an equivalence relation if:

  1. reflexitivity
  2. symmetry
  3. transitivity

Let

s={pqQ(q,7)=1}pqabiff7bpaqs=\{\frac{p}{q}\in Q|(q,7)=1\}\\ \frac{p}{q}\sim\frac{a}{b} iff \frac{7}{bp-aq}

Reflexitivity

For any fraction

ab,abab7abab=70\frac{a}{b}, \frac{a}{b} \sim \frac{a}{b}\\ \frac{7}{ab-ab}=\frac{7}{0} is not reflexitivity


 The relation ~ is not an equivalence relation on s


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