Question

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

The correct option is C

$S$ is an equivalence relation but $R$ is not an equivalence relation

Explanation for the correct options:

Step 1. For the reflexive function, we have

$$\begin{gathered} R:x=wy \\ \left(a,a\right)\in R \end{gathered}$$

Then,

$a=wa$

Then the value of $w$ will be $1$

$\Rightarrow$R is a reflexive

Similarly,

$S=\{\left(\frac{m}{n},\frac{p}{q}\right)$and $qm=pn$

Also,

$\begin{array}{rcl}mn& =& pq\\ & \Rightarrow & qn=pm\end{array}$

$\Rightarrow$$S$ is symmetric.

Step 2. Write the conditions for the function to be equivalence:

if $(ab,ab)\in S$ then $S$ will be reflexive and

If $S\in (ab,cd)and\in (cd,ab)$ then $S$ is symmetric for a function to be transitive

$S\in (2,4),S\in (4,8)$

$\Rightarrow$$S\in (a,b),S\in (4,8)$

$\Rightarrow$$S\in (a,c)$

$\Rightarrow$$S\in (2,8)$

$\Rightarrow$ $S$ is transitive

$\therefore \mathit{S}$ is an equivalence equation.

Hence, option (3) is the correct answer