Answer to Question #153061 in Abstract Algebra for Girish Chandra

Question

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?

Answer 1

The relation is an equivalence relation if:

1. reflexitivity
2. symmetry
3. transitivity

Let

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

Reflexitivity

For any fraction

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

 The relation ~ is not an equivalence relation on s