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Q.
Total number of equivalence relations defined in the set $S = \{a$, $b$, $c\}$ is
Relations and Functions - Part 2
Solution:
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
$R_5 = \{(a$, $a)$, $(b$, $b)$, $(c$, $c)$, $(a$, $b)$, $(b$, $a)$, $(a$, $c)$, $(c$, $a)$, $(b$, $c)$, $(c$, $b)\}$.