Question

The minimum number of elements that must be added to the relation $R=\{(1,2),(2,3)\}$ on the sub set $\{1,2,3\}$ of natural numbers so that it is an equivalence relation is

• A. 4
• B. 7
• C. 6
• D. 5

Answer 1

Answer: (B)

$R$ is reflexive if $(x,x).\in R$ for all $x\in \{1,2,3\}$
$\Rightarrow (1,1),(2,2),(3,3)\in R$
$R$ is symmetric if $(1,2),(2,3)\in R$
$\Rightarrow (2,1),(3,2)\in R\text{. }$
$R$ is transitive if $(1,2),(2,3)\in R$
$\Rightarrow (1,3)\in R$ also $(3,1)\in R$ as $R$ is symmetric.
So the total numbers of elements is 9 .