Question

Mark the correct alternative in the following question:

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

(a) 1 (b) 2 (c) 3 (d) 5

Answer 1

$$\begin{gathered} \mathrm{Consider}\mathrm{the}\mathrm{relation}{R}_1=\left\{\left(1,1\right)\right\} \\ \mathrm{It}\mathrm{is}\mathrm{clearly}\mathrm{reflexive},\mathrm{symmetric}\mathrm{and}\mathrm{transitive} \\ \mathrm{Similarly},R_2=\left\{\left(2,2\right)\right\}\mathrm{and}{R}_3=\left\{\left(3,3\right)\right\}\mathrm{are}\mathrm{reflexive},\mathrm{symmetric}\mathrm{and}\mathrm{transitive} \\ \mathrm{Also},R_4=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right)\right\} \\ \mathrm{It}\mathrm{is}\mathrm{reflexive}\mathrm{as}\left(a,a\right)\in R_4\mathrm{for}\mathrm{all}a\in \left\{1,2,3\right\} \\ \mathrm{It}\mathrm{is}\mathrm{symmetric}\mathrm{as}\left(a,b\right)\in R_4\Rightarrow \left(b,a\right)\in R_4\mathrm{for}\mathrm{all}a\in \left\{1,2,3\right\} \\ \mathrm{Also},\mathrm{it}\mathrm{is}\mathrm{transitive}\mathrm{as}\left(1,2\right)\in R_4,\left(2,1\right)\in R_4\Rightarrow (1,1)\in R_4 \\ \mathrm{The}\mathrm{relation}\mathrm{defined}\mathrm{by}{R}_5=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(1,3\right),\left(2,1\right),\left(2,3\right),\left(3,1\right),\left(3,2\right)\right\}\mathrm{is}\mathrm{reflexive},\mathrm{symmetric}\mathrm{and}\mathrm{transitive}\mathrm{as}\mathrm{well}. \\ \mathrm{Thus},\mathrm{the}\mathrm{maximum}\mathrm{number}\mathrm{of}\mathrm{equivalence}\mathrm{relation}\mathrm{on}\mathrm{set}A=\{1,2,3\}\mathrm{is}5. \end{gathered}$$

Hence, the correct alternative is option (d).