Question

Which one of the following relations on R (set of real numbers) is an equivalence relation

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

Answer 1

Answer: (A)

$a{R}_{1}b\iff \left|a\right|=\left|b\right|$

$a{R}_{1}b⟺|a|=|b|$

$a=b$

$R_1$is a reflexive symmetric and transitive.( as we know $a=b$)

$\therefore (a,a)$ is present.

$\therefore (a,b)(b,a)$ will also be present.

So it is an equivalence relation .