[aditi Chakraborty]

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

[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

 

 

[aditi Chakraborty]

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

[aditi Chakraborty]

Proof of equivalence is :-

Equivalence relation has three critical properties -

1. Reflexive equivalence .eg of its use - A boy is related to himself (he is himself)
2. Symmetric equivalence .eg of his ues - Suman is the friend of mohit, then Mohit is the friend of Suman as well
3. Transitive equivalence. Eg of its use- Ram is the elder brother of Raju , Raju is the elder brother of Raj then Ram is the elder brother of Raj as well.

There are many methods of describing equivalence two of them are -

1. Gluing points is equivalence -

• How we can do this -  declare some points  and glue them together and then visualize in our mind what happens next
• Next we can check this by applying the three critical properties of equivalence-
• Reflexive - If we declare a point A then surely A is related to itself as it is itself A ~A
• Symmetric - If we declare two points A and B and glue them them together then A now becomes related to B as it becomes  B and B becomes A . A~B
• Transitive - If we declare three points A, B and C and glue them together then A becomes related to  B and C as it becomes B and C , B becomes A and C and C becomes A and B. A~B B~C thus A~C

Hence gluing points is an equivalence.

2. Congruence of numbers is an equivalence-

• We can prove this by lets say a number line we declare some points on the number line and check their distance by a  scale  of lets say 10 cm
• Next we can check this by applying the three critical properties of equivalence-
• Reflexive -  If we declare a point A then surely A to A's distance is 0 cm which is divisible by 10 cm. A ~A
• Symmetric -  If we declare two points A and B on the number line and assuming that A B distance is multiples of 10 cm which is divisible by the scale . A ~B
• Transitive - If we declare three points A, B and C on the number line and assuming the distance between A and B ,B and C and A and C are multiples of 10 cm which is divisible by the scale. A~B B~C thus A~C

Thus  congruence of numbers is an equivalence theorem.

PROVED 

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