Equivalence Relation to Partition

[Gouri Basak]

It is true that every equivalence relation gives a partition

For an equivalence relation , the relation must be reflexive,symmetric and transitive

All partitions are reflexive,symmetric and transitive, so it is an equivalence relation.And the equivalence classes of a partition are the sets.So every equivalence relation partitions its set into equivalence classes.

[Saikrish Kailash]

Equivalence Relation guarantees Partition derivation but reverse is not true.

[Shivani Ramesh]

It is true because:

every equivalence relation is

-symmetric

-reflexive and

-transitive

so when we partition the sets they are reflexive, symmetric and transitive

 

[Gouri Basak]

No it is not true that equivalence relation makes a partition and Partition makes an equivalence relation always.

Let's take a way of changing an equivalence relation to a partition first

And for an example let's make a rule that take two numbers x and y such that x and y is divisible by 3,then it is an equivalence relation

Like this many Sets (partitions) can be formed
{0,3,6,9,12....}
{1,4,7,10,13...}

{2,5,8,11,14...}

But these partitions are going infinitely and these are disjointed and exhausted (and can't be brought back to an equivalence relation)

So this example proves that the statement is false .

 

[aditi Chakraborty]

EQUIVALENCE TO PARTITONs

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-If A1~A3(A3 ~ A1 as well) then we can arrange them like {A1,A3} (or like {A3,A1})
• REFLEXIVE – If A2~A2 then we can arrange it like {A2}
• TRANSITIVE– If A4~A5 , A5~A6 and A4 ~A6 (so A6~A5 , A6 ~ A4 and A5 ~ A4 as well)then we can arrange them like {A4,A5,A6} (or like {A6,A5,A4} or {A6,A4,A5} or {A4,A6,A5} or {A5,A6,A4} or {A5,A4,A6})

So,  we can get partitions from equivalence relations.

PARTITIONS TO EQUIVALENCE

We can declare the points in a partition to be equivalent to each other like - suppose we have a set R which has 6 points/elements - {B1,B2,B3,B4,B5,B6}

• SYMMETRIC- If {B1,B2}({B2,B1} Can also be) is a partition then we can declare B1 ~ B2 (so B2 ~ B1 as well)
• REFLEXIVE- If {B3} is a partition then we can declare B3~B3
• TRANSITIVE- If {B4,B5,B6} (or {B4,B6,B5} or {B5,B4,B6} or {B5,B6,B4} or {B6,B4,B5} or {B6,B5,B4}) is a partition then we can declare B4~B5 B5~B6 and B4~B6 ( or B4~B6 B6~B5 and B4~B5 or B5~B6 B6~B4 and B5~B4 or B5~B4 B4~B6 and B6~B5 or B6~B5 B5~B4 and B6~B4 or B6~B4 B4~B5 and B6~B5)

So,  we can get equivalence relations from partitions.

   PROVED

 

 

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