Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $5$

Answer 1

Answer: (D)

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