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Q.
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
Relations and Functions - Part 2
Solution:
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 = \begin{Bmatrix}\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{Bmatrix} $
has 9 elemtns. Starting from 2 elements, therefore the minimum number of elements to be added is 7.