The number of non-empty equivalence relations | vedclass

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(C) An equivalence relation on a set $A$ corresponds to a partition of the set $A$. The number of equivalence relations on a set with $n$ elements is

Vedclass · Accepted Answer

$5$

Vedclass · Answer

(C) An equivalence relation on a set $A$ corresponds to a partition of the set $A$. The number of equivalence relations on a set with $n$ elements is given by the Bell number $B_n$. For the set $A = \{1, 2, 3\}$,the number of elements is $n = 3$. The partitions of $\{1, 2, 3\}$ are: $1$. $\{\{1\}, \{2\}, \{3\}\}$ (corresponds to the identity relation $R = \{(1,1), (2,2), (3,3)\}$) $2$. $\{\{1, 2\}, \{3\}\}$ (corresponds to $R = \{(1,1), (2,2), (3,3), (1,2), (2,1)\}$) $3$. $\{\{1, 3\}, \{2\}\}$ (corresponds to $R = \{(1,1), (2,2), (3,3), (1,3), (3,1)\}$) $4$. $\{\{2, 3\}, \{1\}\}$ (corresponds to $R = \{(1,1), (2,2), (3,3), (2,3), (3,2)\}$) $5$. $\{\{1, 2, 3\}\}$ (corresponds to the universal relation $R = A 	imes A$) Thus,there are $5$ possible equivalence relations. Since all these are non-empty,the total number is $5$.

The number of non-empty equivalence relations on the set $\{1, 2, 3\}$ is :

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