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)

$5$

Answer 2

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)\\}

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