# Given the relation R={(1,2)(2,3)} on set A={1,2,3} add a minimum number of order pair so that enlarged relation is symmetric ,transitive an d reflexive.

a relation on set $A=\left\{1,2,3\right\}$ is reflexive if for every element $x\in A$ ,$xRx$
if R is reflexive relation , missing ordered pair are  $\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$ .
if we add these , the obtain new relation is $R=\left\{\left(1,2\right),\left(2,3\right),\left(1,1\right),\left(2,2\right),\left(3,3\right)\right\}$

a relation is symmetric if for every
therefore following ordered pair are required to make the relation symmetric:$\left\{\left(2,1\right),\left(3,2\right)\right\}$
if we add these, the obtain new relation is $R=\left\{\left(1,2\right),\left(2,3\right),\left(2,1\right),\left(3,2\right)\right\}$

a relation is transitive if for every
so for making it transitive we must add $\left\{\left(1,3\right)\right\}$
the obtain new relation is $\left\{\left(1,2\right),\left(2,3\right),\left(1,3\right)\right\}$

now if the relation is symmetric , transitive and reflexive;
then new relation $R=\left\{\left(1,1\right),\left(2,2\right),\left(3,3\right),\left(1,2\right),\left(2,1\right)\left(2,3\right),\left(3,2\right),\left(1,3\right),\left(3,1\right)\right\}$

hope this helps you

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