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.

if R is reflexive relation , missing ordered pair are $\left\{\right(1,1),(2,2),(3,3\left)\right\}$ .

if we add these , the obtain new relation is $R=\left\{\right(1,2),(2,3),(1,1),(2,2),(3,3\left)\right\}$

a relation is symmetric if for every $a,b\in A,aRb\Rightarrow bRa$

therefore following ordered pair are required to make the relation symmetric:$\left\{(2,1),(3,2)\right\}$

if we add these, the obtain new relation is $R=\left\{\right(1,2),(2,3),(2,1),(3,2\left)\right\}$

a relation is transitive if for every $a,b,c\in A,(aRbbRc\Rightarrow aRc$

so for making it transitive we must add $\left\{\right(1,3\left)\right\}$

the obtain new relation is $\left\{\right(1,2),(2,3),(1,3\left)\right\}$

now if the relation is symmetric , transitive and reflexive;

then new relation $R=\left\{\right(1,1),(2,2),(3,3),(1,2),(2,1\left)\right(2,3),(3,2),(1,3),(3,1\left)\right\}$

hope this helps you

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