First slide

Relations XII

Question

A relation R on the set of complex numbers defined by $Z_{1} R Z_{2} \Leftrightarrow \frac{Z_{1} - Z_{2}}{Z_{1} + Z_{2}} is real then which of the following is not true$

Moderate

A

R is reflexive
B

R is symmetric
C

R is transitive
D

R is not equivalence

Solution

$\begin{array}{l} Given (z_{1}, z_{2}) \in R \Leftrightarrow \frac{z_{1} - z_{2}}{z_{1} + z_{2}} is real number \\ Clearly R is reflexive, because (z_{1}, z_{1}) \in R, Because \frac{z_{1} - z_{1}}{z_{1} + z_{1}} = 0 is a real number \\ Clearly R is symmetric because (z_{1}, z_{2}) \in R \Leftrightarrow (z_{2}, z_{1}) \in R \\ R is transitive also because (z_{1}, z_{2}), (z_{2}, z_{3}) \in R \Rightarrow (z_{1}, z_{3}) \in R \end{array}$
Therefore, the relation is equivalence relation
The correction option is fourth option

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