Question

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

• A. $4$
• B. $7$
• C. $6$
• D. $5$

Answer 1

Answer: (B)

$7$

For $R$ to be equivalence relation, it should be reflexive, symmetric and transitive.

Now, for $R$ reflexive:

$\Rightarrow (x,x)\in R$ for all $x\in \{1,2,3\}$

$\Rightarrow (1,1),(2,2),(3,3)\in R$

For $R$ is symmetric:

$\Rightarrow (1,2),(2,3)\in R$

$\Rightarrow (2,1),(3,2)\in R$

For $R$ is transitive:

$\Rightarrow (1,2),(2,3)\in R$

$\Rightarrow (1,3)\in R$

Also $(3,1)\in R$ as $R$ is symmetric.

So, the total number of elements is $9$.

Hence, minimum $7$ elements must be added.