Question

Let $A=\{a,b,c\}$ and the relation $R$ be defined on $A$ as follows $R=\{(a,a),(b,c),(a,b)\}$. Then write minimum number of ordered pairs to be added in $R$ to make it reflexive and transitive.

Answer 1

Given the relation the relation $R$ be defined on $A$ as follows $R=\{(a,a),(b,c),(a,b)\}$.

Now, to make this relation $R$ to be reflexive we are to add the elements like $(x,x)$ for all $x\in A$.

The the relation will be $R=\left\{\right(a,a),(b,b),(c,c),(b,c),(a,b\left)\right\}$.

Now this relation is reflexive but not transitive as $(a,b)\in A,(b,c)\in A\Rightarrow (a,c)\notin A$.

Now to make this relation to be transitive we are to add $(a,c)$.

Then the relation will be $R=\left\{\right(a,a),(b,b),(c,c),(b,c),(a,b),(a,c\left)\right\}$