A) R is an equivalence relation but S is not an equivalence relation
B) Neither R nor S is an equivalence relation
C) S is an equivalence relation but R is not an equivalence relation
D) R and S both are equivalence relations
Correct Answer: C
Solution :
Probable part of R is {(0, 1), (0, 2)} But \[(1,\text{ }0)\notin R\] as \[1=(w)\text{ }0\] So not symmetric ie. not equivalence Relation \[\frac{m}{n}S\frac{p}{q}\to qm=pn\] Reflexive \[\frac{m}{n}S\frac{p}{q}\to mm=mn\] hence function reflexive . Let \[\frac{m}{n}S\frac{p}{q}\to qm=pn\] Then \[\frac{p}{q}S\frac{m}{n}\to qn=mq\] hence function symmetric \[\frac{m}{s}S\frac{p}{q}\to mq=pn\] (1) \[\frac{p}{q}S\frac{r}{s}\to ps=qr\] (2) eqn. (1)/(2) \[\frac{m}{n}=\frac{r}{s}\to \frac{m}{n}S\frac{r}{s}\] hence transitive So S is equivalence relation
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