Question

Consider the relations \[R = \{ (x,y)\,|\,x,y{\text{ are real numbers and }}x = wy{\text{ for some rational number }}w\} \] and $S = \left\{ {\left( {\dfrac{m}{n},\dfrac{p}{q}} \right)\,|\,m,n,p{\text{ and }}q{\text{ are integers such that }}n,\,q\,{\text{not equal to 0 and }}qm = pn} \right\}$. Then
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

Answer 1

Hint: A relation P is said to be an equivalence relation if P is a reflexive relation, symmetric relation and a transitive relation. Check if the relations R and S are equivalence relations by checking whether they are reflexive, symmetric and transitive.

Complete step by step solution: 
 We know that $R = \{ (x,y)\,|\,x,y{\text{ are real numbers and }}x = wy{\text{ for some rational number }}w\} $.
$(x,x) \in R\,{\text{for }}w = 1$. Therefore, R is reflexive.
$(0,c) \in R\,{\text{for }}w = 0$ but $(c,0) \notin R\,{\text{for any }}c \ne 0$. Therefore, R is not symmetric.
Since R is not symmetric, it cannot be an equivalence relation.
We know that $S = \left\{ {\left( {\dfrac{m}{n},\dfrac{p}{q}} \right)\,|\,m,n,p{\text{ and }}q{\text{ are integers such that }}n,\,q\,{\text{not equal to 0 and }}qm = pn} \right\}$
\[\left( {\dfrac{m}{n},\dfrac{m}{n}} \right) \in S\]as $nm = mn$. Therefore, S is reflexive.
If $\left( {\dfrac{m}{n},\dfrac{p}{q}} \right) \in S$ then $qm = pn$. If $qm = pn$ then $np = mq$. This means that $\left( {\dfrac{p}{q},\dfrac{m}{n}} \right) \in S$. Therefore, S is symmetric.
If $\left( {\dfrac{m}{n},\dfrac{p}{q}} \right) \in S$ and $\left( {\dfrac{p}{q},\dfrac{k}{l}} \right) \in S$ then \[qm = pn\,\] and \[lp = kq\].
From this we can say that$\dfrac{n}{m} = \dfrac{q}{p} = \dfrac{l}{k}$. Therefore, $lm = kn$.
Since $lm = kn$,$\left( {\dfrac{m}{n},\dfrac{k}{l}} \right) \in S$. Therefore, S is transitive.
Since, S is reflexive, symmetric and transitive, S is an equivalence relation.
Therefore, (C) S is an equivalence relation but R is not an equivalence relation is the correct option.

Note: A relation is said to be reflexive if every element of the set is related to itself. A relation is said to be symmetric if for every a related to b, b is also related to a. A relation is said to be transitive if for every a related to b and b related to c, a is also related to c.