[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 .

 

[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.

[Gouri Basak]

To prove - congruence of numbers is an equivalence relation

Let us take a line segment AB of length n and a scale of 3 m to measure it

In this way we can easily prove it when it is reflexive,symmetric and transitive.

i) It is reflexive if we take the point A itself,it measures 0 m and 0 is multiple of every number

ii) It is symmetric because if we take the length between point A and B and if it is divisible by 3, then it is obviously true that the length between point B and A is also divisible by 3

iii) It is also transitive because if we make an extra point C outside AB and the length AB is divisible by 3 and the length BC is divisible by 3 then it is also true that AC is divisible by 3

Thus it proves that congruency of numbers is an equivalence relation

 

[Gouri Basak]

Now we have to prove that :

1- Gluing points is equivalence

2-Congruence is equivalence

Now to just prove that some fact is equivalence,we just need to check it if it is reflexive,symmetric and transitive.

Proof 1 - Gluing points is equivalence

We have to see that gluing points are reflexive,symmetric and transitive or not.(If they aren't then it already tells us that gluing points is not equivalence)

i) It is reflexive as any point A glued to itself is equal

ii) It is symmetric as if any point A is glued to B it means the same when Point B is glued to A

iii) It is also transitive as if any point A is glued to B and that is glued to C it also means the same that Point C is glued to Point A

So it proves that gluing points is equivalence.

Proof 2- Congruence is equivalence

Like in the previous proof,we just have to check if it is reflexive,symmetric and transitive

i) It is reflexive as any object is congruent to itself

ii) It is symmetric as when an object A is congruent to object B then it means the same when the object B is congruent to object A

iii) It is transitive as when an object A is congruent to object B and that is congruent to object C then it also means the same that object C is congruent to object A

So it also proves that congruence is an equivalence.

 

[Author]

Posts