S / R means partition of the given set S into equivalence classes with respect to the equivalence relation R.
We say two elements will belong to the same equivalence class if their R-relative sets are exactly the same.Now let us know what is R-relative set of an element.
An r relative set of an element x is a set of elements y such that x is related to y i.e. x R y
So we find r relative set of each element 1 , 2 , 3 using the above definition.
r(1) = { 1 , 2 } [ As there are ordered pairs (1,1) and (1,2) in the equivalence relation R ]
r(2) = { 1 , 2 } [ As there are ordered pairs (2,1) and (2,2) in the equivalence relation R ]
r(3) = { 3 } [ As there are ordered pairs (3,3) in the equivalence relation R ]
So we see r(1) = r(2) , so one equivalence class will be : [1,2]
And the remaining element 3 will remain separate in its own equivalence class : [3]
So the partition of S with respect to R i.e. quotient S / R = { [1,2] , [3] }
But none of the options matches..So the options are given wrong..
Equivalence partitioning of a set under the corresponding equivalence relation finds application in finding Myhill Nerode equivalence classes which are nothing but the states of the minimised DFA.
