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Q.
The maximum number of equivalence relations on the set $A = \{1$, $2$, $3\}$ are
Relations and Functions - Part 2
Solution:
The smallest equivalence relation is the identity relation $R_1 = \{(1$, $1)$, $(2$, $2)$, $(3$, $3)\}$
Then, two ordered pairs of two distinct elements can be added to give three more equivalence relations.
$R_2 = \{(1$, $1)$, $(2$, $2)$, $(3$, $3)$, $(1$, $2)$, $(2$, $1)\}$
Similarly $R_3$ and $R_4$.
Finally the largest equivalence relation, that is the universal relation.
$R_5 = \{(1$, $1)$, $(2$, $2)$, $(3$, $3)$, $(1$, $2)$, $(2$, $1)$, $(1$, $3)$, $(3$, $1)$, $(2$, $3)$, $(3$, $2)\}$