Question

The maximum number of equivalence relations on the set $A=\{1,2,3\}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $5$

Answer 1

The correct option is D $5$

$R_1=\{(1,1),(2,2),(3,3)\}$

$R_2=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$

$R_3=\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$

$R_4=\{(1,1),(2,2),(3,3),(2,3),(3,2)\}$

$R_5=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}$

These are the $5$ relations on $A$ which are equivalence.