 # Equivalence Relation to Partition

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• #24702

No this is not true that from equivalence relations we can get partitions and from partitions we can get equivalence relations.

There is a simple reason why-

If we make a rule in which we declare that the difference between any two numbers X and Y is divisible by 2 then it is an equivalence relation.

By this  Sets can be formed which can be divided into two partitions

like in set S which contains infinitely many  numbers according to the rule –  {0,2,4,6,8,10,12,14…..}

By this we cant partition all numbers nor get the equivalence relation back from it so we cant get partitions from equivalence relations and equivalence relations from partitions.

So the statement is false.

PROVED

#24706

No the statement is false.

Let’s take an example,

if we make a rule and declare that the difference between x and y is  divisible by 3 which makes an equivalence relation

so in this way we can partition many sets like

{0,3,6,9,12……………}

{1,4,7………………}

{2,5,8,11…………..}

by this we cannot partition all the numbers because it is infinite

and we can get the equivalence relation back so this proves that the statement is false.

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