The relation is an equivalence relation if:
Let
s={pq∈Q∣(q,7)=1}pq∼abiff7bp−aqs=\{\frac{p}{q}\in Q|(q,7)=1\}\\ \frac{p}{q}\sim\frac{a}{b} iff \frac{7}{bp-aq}s={qp∈Q∣(q,7)=1}qp∼baiffbp−aq7
Reflexitivity
For any fraction
ab,ab∼ab7ab−ab=70\frac{a}{b}, \frac{a}{b} \sim \frac{a}{b}\\ \frac{7}{ab-ab}=\frac{7}{0}ba,ba∼baab−ab7=07 is not reflexitivity
The relation ~ is not an equivalence relation on s
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