Question

Given the relation r = {(1, 2), (2, 3)} on the set a = {1, 2, 3}, the minimum number of ordered pairs which when added to r make it an equivalence relation is

Answer 1

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

a relation is symmetric if for every a,b∈A ,aRb⇒bRa
therefore following ordered pair are required to make the relation symmetric:{(2,1),(3,2)}
if we add these, the obtain new relation is R={(1,2),(2,3),(2,1),(3,2)}

a relation is transitive if for every a,b,c∈A, (aRb &bRc ⇒aRc
so for making it transitive we must add {(1,3)}
the obtain new relation is {(1,2),(2,3),(1,3)}

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