Equivalence Relation to Partition

Home Forums Math Olympiad, I.S.I., C.M.I. Entrance Combinatorics Equivalence Relation to Partition

Viewing 5 posts - 1 through 5 (of 12 total)
  • Author
  • #24686
    Ashani Dasgupta

    Does every equivalence relation defined on a set, automatically partitions the set?

    If yes, give a rigorous proof.

    If no, give an example.

    Pinaki Biswas

    Proof:1. If there is x in a set, then it will have to be in a partition.(symmetric)

    2.If x ~y, they are in the same partition.


    3.If x~y,y~z,

    Then x~z as all are in the same partition.(transitive)



    We can partition the points/elements according to their equivalence like – suppose we have a set S which has 6 points/elements- {A1,A2,A3,A4,A5,A6}

    • SYMMETRIC-A1~A3 then we can arrange them like {A1,A3}
    • REFLEXIVE – A2~A2 then we can arrange it like {A2}
    • TRANSITIVE– A4~A5 , A5~A6 and A4 ~A6 then we can arrange them like {A4,A5,A6}

    So,  we can get partitions from equivalence relations


    Harshit Shah


    Equivalence relation -> Partition

    Constrtuct partitions from the relation as follows,

    1) pick an element from set S, take all y in S such that x~y and put them all in a new partition

    2) repeat step 1 until all elements of S are exhausted

    step 1 guarantees disjointedness and step 2 guarantees exhaustiveness


    Partition -> Equivalence relation

    Declare all elements in a single partition to be equivalent,

    1) reflexivity: trivial (x is in the same partition as x, so x~x)

    2) symmetry: if x is in the same partition as y, so y is also in the same partition. Therefore, x~y=>y~x

    3) transitivity: if x is in the same partition as y, y is in the same partition as z, then x is in the same partition as z. Therefore, x~y,y~z => x~z





    • This reply was modified 1 year, 8 months ago by Harshit Shah.
    Writaban Sarkar

    Yes, since making equivalence relations means that the elements are in the same set, we can say that stating elements equivalent means to break the set and form some new set (partition).

Viewing 5 posts - 1 through 5 (of 12 total)
  • You must be logged in to reply to this topic.