QJEEM10Ma3665: JEE Main 2010 Mathematics - Class 11 JEEM10Ma3665 Solution: Consider the following relations are real numbers and for some…

Let's evaluate each relation for the properties of an equivalence relation: reflexivity, symmetry, and transitivity.

Relation R : $R=(x, y) \mid x, y$ are real numbers and $x=w y$ for some rational number $w$ .

• Reflexivity : For all $x$ in $R$ , $x = 1x$ . Since 1 is a rational number, every element is related to itself.


• Symmetry : For all $x, y$ in $R$ , if $x = w y$ for some rational $w$ , then $y = \frac{1}{w}x$ . However, if $w = 0$ , then $\frac{1}{w}$ is undefined, and therefore, $R$ doesn't satisfy symmetry.


• Transitivity : If $x = w y$ and $y = v z$ for some rational numbers $w$ and $v$ , then $x = (w v)z$ . Since the product of rational numbers is rational, if $x$ is related to $y$ and $y$ is related to $z$ , then $x$ is related to $z$ .

Therefore, $R$ is not an equivalence relation on $R$ since it does not satisfy the symmetry property.

Relation S : $S=\left\{\left(\frac{m}{n}, \frac{p}{q}\right) \mid m, n, p\right.$ and $q$ are integers such that $n, q \neq 0$ and $q m=p m\}$

• Reflexivity : For all $\frac{m}{n}$ , $\frac{m}{n} = \frac{m}{n}$ . Since $n \neq 0$ and $m = m$ , every element is related to itself.


• Symmetry : For all $\frac{m}{n}$ , $\frac{p}{q}$ , if $q m = p n$ , then $n p = m n$ . So if $\frac{m}{n}$ is related to $\frac{p}{q}$ , then $\frac{p}{q}$ is related to $\frac{m}{n}$ .


• Transitivity :
  
  $\begin{array}{rlr}\frac{m}{n} R \frac{p}{q} \text { and } \frac{p}{q} R \frac{r}{s} \\\\\Rightarrow m q=n p \text { and } p s=r q \\\\\Rightarrow m q \cdot p s=n p \cdot r q \\\\\Rightarrow \quad m s=n r \\\\\Rightarrow \frac{m}{n}=\frac{r}{s} \Rightarrow \frac{m}{n} R \frac{r}{s}\end{array}$
  
  So if $\frac{m}{n}$ is related to $\frac{p}{q}$ and $\frac{p}{q}$ is related to $\frac{r}{s}$ , then $\frac{m}{n}$ is related to $\frac{r}{s}$ . The relation $S$ is transitive.

Therefore, $S$ is an equivalence relation on the set of all fractions where denominator is not zero.

In conclusion, the correct answer is

Option C : $S$ is an equivalence relation but $R$ is not an equivalence relation.