Question

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

• A. $3$
• B. $1$
• C. $5$
• D. $2$

Answer 1

The correct option is C $5$

Construct equivalence relations between the elements of set $A.$

Given: $A=\{1,2,3\}$

Here, the equivalence relations are as follows:

$R_1=\left\{\right(1,1),(2,2),(3,3\left)\right\}$

$R_2=\{1,1),(2,2),(3,3),(1,2),(2,1)\}$

$R_3=\left\{\right(1,1),(2,2),(3,3),(1,3),(3,1\left)\right\}$

$R_4=\left\{\right(1,1),(2,2),(3,3),(2,3),(3,2\left)\right\}$

$R_5=$$$\left\{\begin{array}{cc}(1,1),(2,2),(3,3),(1,2),(1,3)& \\ (2,1),(2,3),(3,1),(3,2)& \end{array}\right\}$$
Thus, the maximum number of equivalence relation is ${}^{\prime }{5}^{\prime }$
Hence, corect answer is option (d)