Question

Which one of the following relations on $R$ is an equivalence relation?

• A. $aR_{1}b \Leftrightarrow \left|a\right|=\left|b\right|$
• B. $aR_{2}b \Leftrightarrow a \ge b$
• C. $aR_{3}b \Leftrightarrow a$ divides $b$
• D. $aR_{4}b \Leftrightarrow a < b$

Answer 1

Answer: (A)

$aR_{1}b \Leftrightarrow \left|a\right|=\left|b\right|$

Answer 2

$(i)$ Reflexive : $a \in R$, $aR_1a$ $\Rightarrow |a| = |a|$ $(ii)$ Symmetric : $a$, $b \in R$ $aR_{1}b \Rightarrow \left|a\right| = \left|b\right|$ $\Rightarrow \left|b\right| = \left|a\right|$ $\Rightarrow bR_{1}a$ $(iii)$ Transitive : $a$, $b$, $c \in R$ $aR_{1}b \Rightarrow \left|a\right|=\left|b\right|$, $bR_{1}c$ $\Rightarrow \left|b\right|=\left|c\right|$. So, $\left|a\right|=\left|c\right|$ $\Rightarrow aR_{1}c$ $\Rightarrow R_{1}$ is an equivalence relation on $R$.