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^{nm}relations between them. But my question is, does this count redundancies? What I mean is, if in the relation A~B = B~A. I don't want to count identical relations twice.

Thanks!

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- Thread starter Battlemage!
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Thanks!

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- #4

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Multiplication is not a relation, it is an operation.

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^{nm}relations between them. But my question is, does this count redundancies? What I mean is, if in the relation A~B = B~A. I don't want to count identical relations twice.

Thanks!

A relation between sets A and B is by definition a subset of AxB. If A has n elements and B has m elements, AxB has nm elements. Such a set has 2

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I thought all operations were types of relations. It appears i mixed some stuff up. Lets just rephrase the question as multuplication instead.Multiplication is not a relation, it is an operation.

- #7

Mark44

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So what is the new question?I thought all operations were types of relations. It appears i mixed some stuff up. Lets just rephrase the question as multuplication instead.

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Im sorry I havent slept in 38 hours that last post was nonsense.So what is the new question?

But as a test, a specific example:

Say I have a set with 6 elements and a set with 4 elements. Call them set A=[a1, a2, a3, a4, a5, a6] and set B=[b1,b2,b3,b4] Each of the 6 elements can be matched with one of the 4 elements, but each of the 4 elements can be matched to any number of the 6. So you can end up with all 6 of set A being associated with the first element of set B, or a1 and a2 being mathed woth b1, a3 being matched with b2, a4 and a5 being mathed with b3, and a6 being matched with b4. And so on. Any combination as long as all 6 from set A are matched with elements from set B and

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Stephen Tashi

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Say you have set A with n elements and set B with m elements. If I recall, there are a total of 2^{nm}relations between them.

To investigate the correctness of that statement, we would need an clear definition for the statement "##R## is a relation between set ##A## and set ##B##".

For example, if ##R =\{(1,4), (2,4)\}## then is ##R## a relation between the sets ##A =\{1,2,3\}## and ##B=\{4\}##?

It looks to me like your result of ##2^{mn}## would count ##R## as being among the relations "between" ##A## and ##B##.

However, doesn't ##2^{mn}## also count the null set of ordered pairs as a relation? Are we going to count a null set of ordered pairs as a relation ? Is it a relation "between" ##A## and ##B## ?

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Stephen Tashi

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Are the sets AXB and BXA identical

Yes, if ##A=B##.

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1. The set A is related to set B.

2.The set B is related to set A.

3. There is a relation between the sets A and B.

The definition given by Cruz Martinez favors the third option. There is a relation between A and B means A is related to B and B is related to A.

- #15

Stephen Tashi

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Stephen Tashi, thanks for responding. What is the correct or appropriate statement of teh following:

1. The set A is related to set B.

2.The set B is related to set A.

3. There is a relation between the sets A and B.

I don't know of any standard mathematical definition for a relation "between" sets.

So let's look at your post #8 and consider what you want the statement "##R## is a relation between ##A## and ##B##" to mean.

I think you want each element of ##A## to in at least one of the ordered pairs of ##R##.

With that requirement, we would not consider ##R = \{(a1,b1), (a2,b4)\}## to be a relation between ##A## and ##B## because some of the elements of ##A## are missing in the first members of the ordered pairs of ##R##.

Now lets consider ##R = \{ (a1,b1),(a1,b2), (a3,b4), (a4,b4) ,(a5,b4), (a6,b4)\}##. Do you want ##R## to be a relation between ##A## and ##B## or do you want to disqualify it because it contains both ##(a1,b1)## and ##(a1,b2)## ?

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I don't know of any standard mathematical definition for a relation "between" sets.

I've seen it in some books, e.g. Lee, Topological Manifolds, Appendix A. The other definition I've seen for a relation is that R is a relation if whenever z ∈ R, z is an ordered pair.

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I've seen it in some books, e.g. Lee, Topological Manifolds, Appendix A. The other definition I've seen for a relation is that R is a relation if whenever z ∈ R, z is an ordered pair.

Yes, you're right. But you and Stephen are talking about something else. You are talking about a relation with codomain and domain a set. In this case, two elements of the sets can be related. Stephen seems to be talking about the case where two sets themselves are related. Of course you can define such a relation, but there is no standard one like the OP thinks.

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$$R~\text{ is a relation}~\Leftrightarrow~\forall r(r \in R~\rightarrow~\exists x \exists y(r~=~<x,y>)~)$$

This definition leave open the possibility that some members of relation ##R## are not ordered pairs.

However, the cross product, ##A~\times~B##, of two sets, ##A## and ##B##, is certainly a relation since it satisfies the definition given above.

The minimum number of possible relations between sets ##A## and ##B## would then be the number of sets in the power set of ##A~\times~B##, but the maximum number of relations is infinite.

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This definition leave open the possibility that some members of relation ##R## are not ordered pairs.

Then it is not the standard definition.

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Ok, if you want to be picky :), the definition leaves open the possibility that some elements of the relation ##R## are not ordered pairs.

No. A relation by definition consists only of ordered pairs.

- #23

Stephen Tashi

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You haven't yet stated a definition for "##R## is a relation between sets ##A## and ##B##".

I think the definition you are using implicitly is:

##R## is a relation between sets ##A## and ##B## if and only if ##R## is a subset of ##A \times B##.

That's a reasonable definition for some purposes, but I wouldn't say that its a

To answer the question in the original post, we need to know what definition Battlemage wants to use. Perhaps Battlemage's doesn't really want to know about relations. Perhaps his question is:

"How many distinct functions are there that have a domain consisting of set ##A## with cardinality is ##n## and an image that is a non-null subset of set ##B##, where ##B## has cardinality ##m## ?"

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You haven't yet stated a definition for "##R## is a relation between sets ##A## and ##B##".

I think the definition you are using implicitly is:

##R## is a relation between sets ##A## and ##B## if and only if ##R## is a subset of ##A \times B##.

That's a reasonable definition for some purposes, but I wouldn't say that its astandarddefinition.

To answer the question in the original post, we need to know what definition Battlemage wants to use. Perhaps Battlemage's doesn't really want to know about relations. Perhaps his question is:

"How many distinct functions are there that have a domain consisting of set ##A## with cardinality is ##n## and an image that is a non-null subset of set ##B##, where ##B## has cardinality ##m## ?"

Thank you for asking.

Here is what I'm getting at. I have two sets. The first set A is smaller than the second set B. Each element from A must be linked with one and only one element from B, but any element from B can be linked to as many elements of A as there are in A (or as few as zero).

How many possible configurations are there?

I hesitate to use the word "pair" because in this scenario one element from B can have multiple elements from A associated with it, while each element from A can only have one element from B associated with it.

Also if B were smaller than A but the same rules applied (each element of A must be linked to an elememt of B, while each element of B can be linked with as few as zero elements to as many as are in A.).

I do apologize for not using correct technical verbiage here.

Edited to add: the motivation for this came from me thinking about a training problem i had at my new job. We were to categorize documents by how confidential they were. There were a few categories of confidentiality but many types of documents. Being quickly bored of the exercise I was wondering at how big the possible number of arrangements could be, and how that question could be answered in general. I realize most people would question my sanity over such idle curiosity...

One more edit- my first estimate was 2

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