Question

The minimum number of elemetns that must be added to the relation $R=\{(1,2),(2,3)\}$ 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)

Since $R=\{(1,2),(2,3)\}$
i.e. $A=\{1,2\}$ and $B=\{2,3\}$
Now if $R_1$ is the Reflexive relation, such that
$R_1=\{(1,2),(2,3),(1,1),(2.2).(3,3)\}$ has 5 elements
Now, If $R_2$ is both symmetric & reflexive relation, then
$R_2=\{(1,2),(2,3),(1,1),(2,2),(2,1),(3,2),(3,3)\}$ has 7 elements
Again, R- is reflexive, symmetric and transitive a!i together, then
$R_3=\left\{\begin{array}{ccc}\left(1,2\right)& \left(2,3\right)& \left(1,1\right)\\ \left(2,2\right)& \left(2,1\right)& \left(3,2\right)\\ \left(3,3\right)& \left(1,3\right)& \left(3,1\right)\end{array}\right\}$
has 9 elemtns. Starting from 2 elements, therefore the minimum number of elements to be added is 7.