# Can the size of any E-class be either 9 or 99?

• bedi
In summary: Yes this is an equivalence relation.{(a,a) : a in A} + {({2},{2}), ({2},{3}), ({3},{2}), ({3},{3})}.This is not an equivalence relation.
bedi

## Homework Statement

Let A be the set of all subsets of {1,2,...,99} and E an equivalence
relation on A. Can the size of any E-class be either 9 or 99?

## The Attempt at a Solution

I think yes but can't see how...

How many subsets of A are there?

If you want to define an equivalence relation where each class has 9 members, how many equivalence classes do you get? Does that make sense?

mfb said:
How many subsets of A are there?

If you want to define an equivalence relation where each class has 9 members, how many equivalence classes do you get? Does that make sense?

Much depends on the meaning of "any" in the question "Can the size of any E-class be either 9 or 99?" You seem to be assuming that "any" = "every", but maybe the question wants to know if anyone (but not all) of the equivalence classes can have size 9 (or 99).

I thought about that, but that would make the question weird and (imho) pointless.

The only relevancy that I see, is as an exercise to understand the definitions of equivalence relations and of equivalence classes.

bedi, can you give an example of an equivalence relation on A?
And more to the point, can you give an example of an equivalence relation that has an equivalence class of size 9?

I like Serena said:
bedi, can you give an example of an equivalence relation on A?
And more to the point, can you give an example of an equivalence relation that has an equivalence class of size 9?

As an example of an equivalence relation on A:
Provided X,Y are elements of A, X≈Y iff either X is a subset of Y or Y is a subset of X. In view of this relation ≈ every element of A is related, right?
But I cannot find an equivalence relation that has an equivalence class of size 9... And cannot see why it is impossible.

Good. That is indeed an equivalence relation.
One that has only 1 equivalence class, which is A.

What about the relation given by the set of pairs:
{(a,a) : a in A} + {({2},{2}), ({2},{3}), ({3},{2}), ({3},{3})}.

Is it an equivalence relation on A?
If so which equivalence classes does it contain?

Last edited:
Yes I think this is an equivalence relation on A. And the equivalence classes are:
[1]={{1}}, [2]={{2},{3}}, [3]={{3}}. Actually I'm not sure if an equivalence relation on any given set must say something about all the elements of the set. For instance in this eq. relation we do not have [4], is this a problem?

Actually, you'd have [3]={{2},{3}}.
And so {{2},{3}} is the corresponding equivalence class of size 2.

From wiki:

A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive.
Equivalently, for all a, b and c in A:
1. a ~ a. (Reflexivity)
2. if a ~ b then b ~ a. (Symmetry)
3. if a ~ b and b ~ c then a ~ c. (Transitivity)

So yes, each element a in A must at least have an equivalence relation with itself.Now, can you think of an equivalence relation on A that contains an equivalence class of size 9?

Last edited:
On the other hand, mfb asked that "if you want to define an equivalence relation where each class has 9 members, how many equivalence classes do you get?" and since equivalence classes are also a partition on A, we have to have 2^99/9 classes in this case, which is not possible. Correct?

bedi said:
On the other hand, mfb asked that "if you want to define an equivalence relation where each class has 9 members, how many equivalence classes do you get?" and since equivalence classes are also a partition on A, we have to have 2^99/9 classes in this case, which is not possible. Correct?

Correct. ;)

Thank you, you are so helpful!

Btw, I concur with Ray that this is not what the problem asks.

The problem asks for "any", and not for "every" or "each".
I interpret it that it suffices if there is just a single equivalence class of size 9.

Alright, that can happen right? Why not?

To properly answer the question, you have to give a specific example, and prove that it is indeed an equivalence relation.

But yeah, it can happen. ;)

Well, it is trivial.
Take any 9 elements of A, let them be in the same equivalence class, ignore the other elements (apart from the required M~M for every element of A).
It is not an interesting equivalence relation, but that was not requested.

bedi said:
As an example of an equivalence relation on A:
Provided X,Y are elements of A, X≈Y iff either X is a subset of Y or Y is a subset of X. In view of this relation ≈ every element of A is related, right?
I don't see how this is an equivalence relation.

{1,2}~{1}~{1,3}, but not {1,2}~{1,3}

## 1. Can the size of any E-class be 9 or 99?

Yes, the size of any E-class can be either 9 or 99. This is because the E-class is a general classification for a certain type of object or group, and the specific size can vary within that classification.

## 2. Are there any exceptions to the E-class size being either 9 or 99?

There may be some rare exceptions where the size of an E-class object may not be exactly 9 or 99, but these would be outliers and not representative of the typical size range for this classification.

## 3. Why are the size options for E-class limited to just 9 or 99?

The size options for E-class are not necessarily limited to just 9 or 99, but these are commonly used sizes that have been established within the classification system. Other sizes may also exist, but they may not be as widely recognized or used.

## 4. Can the size of an E-class change from 9 to 99 or vice versa?

Yes, the size of an E-class can change from 9 to 99 or vice versa. This could happen if there is a change in the classification system or if new information is discovered about the object or group that warrants a different size designation.

## 5. How is the size of an E-class determined?

The size of an E-class is typically determined by the characteristics and features of the object or group it represents. This can include measurements, observations, and comparisons to other objects within the same classification system.

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