Equivalence Relation to Partition

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• #24686

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

If yes, give a rigorous proof.

If no, give an example.

#24687

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.

=>y~x(reflexive)

3.If x~y,y~z,

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

#24689

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

PROVED

#24690

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

#24691

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).

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