Question

Total number of equivalence relations defined in the set $S=\{a$, $b$, $c\}$ is

• A. $5$
• B. $3!$
• C. $23$
• D. $33$

Answer 1

Answer: (A)

The smallest equivalence relation is the identity relation $R_1=\{(a$ , $a)$, $(b$, $b)$, $(c$, $c)\}$
Then, two ordered pairs of two distinct elements can be added to give three more equivalence relations
$R_2=\{(a$, $a)$, $(b$, $b)$, $(c$, $c)$, $(a$, $b)$, $(b$, $a)\}$
Similarly $R_3$ and $R_4$.
Finally, the largest equivalence relation, that is the universal relation
$R_5=\{(a$, $a)$, $(b$, $b)$, $(c$, $c)$, $(a$, $b)$, $(b$, $a)$, $(a$, $c)$, $(c$, $a)$, $(b$, $c)$, $(c$, $b)\}$.